acos(((x + y + z) - 1)/2)

Percentage Accurate: 100.0% → 100.0%
Time: 4.9s
Alternatives: 7
Speedup: 1.0×

Specification

?
\[\left(\left(-1 \leq x \land x \leq 1\right) \land \left(-1 \leq y \land y \leq 1\right)\right) \land \left(-1 \leq z \land z \leq 1\right)\]
\[\begin{array}{l} \\ \cos^{-1} \left(\frac{\left(\left(x + y\right) + z\right) - 1}{2}\right) \end{array} \]
(FPCore (x y z) :precision binary64 (acos (/ (- (+ (+ x y) z) 1.0) 2.0)))
double code(double x, double y, double z) {
	return acos(((((x + y) + z) - 1.0) / 2.0));
}

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = acos(((((x + y) + z) - 1.0d0) / 2.0d0))
end function

public static double code(double x, double y, double z) {
	return Math.acos(((((x + y) + z) - 1.0) / 2.0));
}

def code(x, y, z):
	return math.acos(((((x + y) + z) - 1.0) / 2.0))

function code(x, y, z)
	return acos(Float64(Float64(Float64(Float64(x + y) + z) - 1.0) / 2.0))
end

function tmp = code(x, y, z)
	tmp = acos(((((x + y) + z) - 1.0) / 2.0));
end

code[x_, y_, z_] := N[ArcCos[N[(N[(N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\cos^{-1} \left(\frac{\left(\left(x + y\right) + z\right) - 1}{2}\right)
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 7 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{\left(\left(x + y\right) + z\right) - 1}{2}\right) \end{array} \]
(FPCore (x y z) :precision binary64 (acos (/ (- (+ (+ x y) z) 1.0) 2.0)))
double code(double x, double y, double z) {
	return acos(((((x + y) + z) - 1.0) / 2.0));
}

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = acos(((((x + y) + z) - 1.0d0) / 2.0d0))
end function

public static double code(double x, double y, double z) {
	return Math.acos(((((x + y) + z) - 1.0) / 2.0));
}

def code(x, y, z):
	return math.acos(((((x + y) + z) - 1.0) / 2.0))

function code(x, y, z)
	return acos(Float64(Float64(Float64(Float64(x + y) + z) - 1.0) / 2.0))
end

function tmp = code(x, y, z)
	tmp = acos(((((x + y) + z) - 1.0) / 2.0));
end

code[x_, y_, z_] := N[ArcCos[N[(N[(N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\cos^{-1} \left(\frac{\left(\left(x + y\right) + z\right) - 1}{2}\right)
\end{array}

Alternative 1: 100.0% accurate, 0.9× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + z\right)\\ \cos^{-1} \left(\frac{\mathsf{fma}\left(t_0, t_0, -1\right)}{\mathsf{fma}\left(t_0, 2, 2\right)}\right) \end{array} \end{array} \]
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (+ y z)))) (acos (/ (fma t_0 t_0 -1.0) (fma t_0 2.0 2.0)))))
double code(double x, double y, double z) {
	double t_0 = x + (y + z);
	return acos((fma(t_0, t_0, -1.0) / fma(t_0, 2.0, 2.0)));
}

function code(x, y, z)
	t_0 = Float64(x + Float64(y + z))
	return acos(Float64(fma(t_0, t_0, -1.0) / fma(t_0, 2.0, 2.0)))
end

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y + z), $MachinePrecision]), $MachinePrecision]}, N[ArcCos[N[(N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision] / N[(t$95$0 * 2.0 + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := x + \left(y + z\right)\\
\cos^{-1} \left(\frac{\mathsf{fma}\left(t_0, t_0, -1\right)}{\mathsf{fma}\left(t_0, 2, 2\right)}\right)
\end{array}
\end{array}
Derivation

  1. Initial program 100.0%

    \[\cos^{-1} \left(\frac{\left(\left(x + y\right) + z\right) - 1}{2}\right) \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. flip--N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\frac{\left(\left(x + y\right) + z\right) \cdot \left(\left(x + y\right) + z\right) - 1 \cdot 1}{\left(\left(x + y\right) + z\right) + 1}}}{2}\right) \]

    2. associate-/l/N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{\left(\left(x + y\right) + z\right) \cdot \left(\left(x + y\right) + z\right) - 1 \cdot 1}{2 \cdot \left(\left(\left(x + y\right) + z\right) + 1\right)}\right)} \]

    3. /-lowering-/.f64N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{\left(\left(x + y\right) + z\right) \cdot \left(\left(x + y\right) + z\right) - 1 \cdot 1}{2 \cdot \left(\left(\left(x + y\right) + z\right) + 1\right)}\right)} \]

    4. metadata-evalN/A

      \[\leadsto \cos^{-1} \left(\frac{\left(\left(x + y\right) + z\right) \cdot \left(\left(x + y\right) + z\right) - \color{blue}{1}}{2 \cdot \left(\left(\left(x + y\right) + z\right) + 1\right)}\right) \]

    5. sub-negN/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\left(\left(x + y\right) + z\right) \cdot \left(\left(x + y\right) + z\right) + \left(\mathsf{neg}\left(1\right)\right)}}{2 \cdot \left(\left(\left(x + y\right) + z\right) + 1\right)}\right) \]

    6. accelerator-lowering-fma.f64N/A

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\mathsf{fma}\left(\left(x + y\right) + z, \left(x + y\right) + z, \mathsf{neg}\left(1\right)\right)}}{2 \cdot \left(\left(\left(x + y\right) + z\right) + 1\right)}\right) \]

    7. associate-+l+N/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(\color{blue}{x + \left(y + z\right)}, \left(x + y\right) + z, \mathsf{neg}\left(1\right)\right)}{2 \cdot \left(\left(\left(x + y\right) + z\right) + 1\right)}\right) \]

    8. +-lowering-+.f64N/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(\color{blue}{x + \left(y + z\right)}, \left(x + y\right) + z, \mathsf{neg}\left(1\right)\right)}{2 \cdot \left(\left(\left(x + y\right) + z\right) + 1\right)}\right) \]

    9. +-lowering-+.f64N/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(x + \color{blue}{\left(y + z\right)}, \left(x + y\right) + z, \mathsf{neg}\left(1\right)\right)}{2 \cdot \left(\left(\left(x + y\right) + z\right) + 1\right)}\right) \]

    10. associate-+l+N/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(x + \left(y + z\right), \color{blue}{x + \left(y + z\right)}, \mathsf{neg}\left(1\right)\right)}{2 \cdot \left(\left(\left(x + y\right) + z\right) + 1\right)}\right) \]

    11. +-lowering-+.f64N/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(x + \left(y + z\right), \color{blue}{x + \left(y + z\right)}, \mathsf{neg}\left(1\right)\right)}{2 \cdot \left(\left(\left(x + y\right) + z\right) + 1\right)}\right) \]

    12. +-lowering-+.f64N/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(x + \left(y + z\right), x + \color{blue}{\left(y + z\right)}, \mathsf{neg}\left(1\right)\right)}{2 \cdot \left(\left(\left(x + y\right) + z\right) + 1\right)}\right) \]

    13. metadata-evalN/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(x + \left(y + z\right), x + \left(y + z\right), \color{blue}{-1}\right)}{2 \cdot \left(\left(\left(x + y\right) + z\right) + 1\right)}\right) \]

    14. distribute-rgt-inN/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(x + \left(y + z\right), x + \left(y + z\right), -1\right)}{\color{blue}{\left(\left(x + y\right) + z\right) \cdot 2 + 1 \cdot 2}}\right) \]

    15. metadata-evalN/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(x + \left(y + z\right), x + \left(y + z\right), -1\right)}{\left(\left(x + y\right) + z\right) \cdot 2 + \color{blue}{2}}\right) \]

    16. accelerator-lowering-fma.f64N/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(x + \left(y + z\right), x + \left(y + z\right), -1\right)}{\color{blue}{\mathsf{fma}\left(\left(x + y\right) + z, 2, 2\right)}}\right) \]

    17. associate-+l+N/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(x + \left(y + z\right), x + \left(y + z\right), -1\right)}{\mathsf{fma}\left(\color{blue}{x + \left(y + z\right)}, 2, 2\right)}\right) \]

    18. +-lowering-+.f64N/A

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(x + \left(y + z\right), x + \left(y + z\right), -1\right)}{\mathsf{fma}\left(\color{blue}{x + \left(y + z\right)}, 2, 2\right)}\right) \]

    19. +-lowering-+.f64100.0

      \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(x + \left(y + z\right), x + \left(y + z\right), -1\right)}{\mathsf{fma}\left(x + \color{blue}{\left(y + z\right)}, 2, 2\right)}\right) \]

  4. Applied egg-rr100.0%

    \[\leadsto \cos^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(x + \left(y + z\right), x + \left(y + z\right), -1\right)}{\mathsf{fma}\left(x + \left(y + z\right), 2, 2\right)}\right)} \]
  5. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{\left(z + \left(x + y\right)\right) + -1}{2}\right) \end{array} \]
(FPCore (x y z) :precision binary64 (acos (/ (+ (+ z (+ x y)) -1.0) 2.0)))
double code(double x, double y, double z) {
	return acos((((z + (x + y)) + -1.0) / 2.0));
}

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = acos((((z + (x + y)) + (-1.0d0)) / 2.0d0))
end function

public static double code(double x, double y, double z) {
	return Math.acos((((z + (x + y)) + -1.0) / 2.0));
}

def code(x, y, z):
	return math.acos((((z + (x + y)) + -1.0) / 2.0))

function code(x, y, z)
	return acos(Float64(Float64(Float64(z + Float64(x + y)) + -1.0) / 2.0))
end

function tmp = code(x, y, z)
	tmp = acos((((z + (x + y)) + -1.0) / 2.0));
end

code[x_, y_, z_] := N[ArcCos[N[(N[(N[(z + N[(x + y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\cos^{-1} \left(\frac{\left(z + \left(x + y\right)\right) + -1}{2}\right)
\end{array}
Derivation

  1. Initial program 100.0%

    \[\cos^{-1} \left(\frac{\left(\left(x + y\right) + z\right) - 1}{2}\right) \]
  2. Add Preprocessing

  3. Final simplification100.0%

    \[\leadsto \cos^{-1} \left(\frac{\left(z + \left(x + y\right)\right) + -1}{2}\right) \]
  4. Add Preprocessing

Alternative 3: 98.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -4 \cdot 10^{-15}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(x, 0.5, -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(0.5, y + z, -0.5\right)\right)\\ \end{array} \end{array} \]
(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -4e-15)
   (acos (fma x 0.5 -0.5))
   (acos (fma 0.5 (+ y z) -0.5))))
double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -4e-15) {
		tmp = acos(fma(x, 0.5, -0.5));
	} else {
		tmp = acos(fma(0.5, (y + z), -0.5));
	}
	return tmp;
}

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -4e-15)
		tmp = acos(fma(x, 0.5, -0.5));
	else
		tmp = acos(fma(0.5, Float64(y + z), -0.5));
	end
	return tmp
end

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -4e-15], N[ArcCos[N[(x * 0.5 + -0.5), $MachinePrecision]], $MachinePrecision], N[ArcCos[N[(0.5 * N[(y + z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x + y \leq -4 \cdot 10^{-15}:\\
\;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(x, 0.5, -0.5\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(0.5, y + z, -0.5\right)\right)\\


\end{array}
\end{array}
Derivation
  1. Split input into 2 regimes

  2. if (+.f64 x y) < -4.0000000000000003e-15

    1. Initial program 99.7%

      \[\cos^{-1} \left(\frac{\left(\left(x + y\right) + z\right) - 1}{2}\right) \]
    2. Add Preprocessing

    3. Taylor expanded in y around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot \left(\left(x + z\right) - 1\right)\right)} \]
    4. Step-by-step derivation

      1. sub-negN/A

        \[\leadsto \cos^{-1} \left(\frac{1}{2} \cdot \color{blue}{\left(\left(x + z\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      2. metadata-evalN/A

        \[\leadsto \cos^{-1} \left(\frac{1}{2} \cdot \left(\left(x + z\right) + \color{blue}{-1}\right)\right) \]

      3. distribute-lft-inN/A

        \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot \left(x + z\right) + \frac{1}{2} \cdot -1\right)} \]

      4. metadata-evalN/A

        \[\leadsto \cos^{-1} \left(\frac{1}{2} \cdot \left(x + z\right) + \color{blue}{\frac{-1}{2}}\right) \]

      5. accelerator-lowering-fma.f64N/A

        \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, x + z, \frac{-1}{2}\right)\right)} \]

      6. +-commutativeN/A

        \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{z + x}, \frac{-1}{2}\right)\right) \]

      7. +-lowering-+.f6483.0

        \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(0.5, \color{blue}{z + x}, -0.5\right)\right) \]

    5. Simplified83.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(0.5, z + x, -0.5\right)\right)} \]

    6. Taylor expanded in z around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot x - \frac{1}{2}\right)} \]
    7. Step-by-step derivation

      1. sub-negN/A

        \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

      2. *-commutativeN/A

        \[\leadsto \cos^{-1} \left(\color{blue}{x \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      3. metadata-evalN/A

        \[\leadsto \cos^{-1} \left(x \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}\right) \]

      4. accelerator-lowering-fma.f6481.8

        \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(x, 0.5, -0.5\right)\right)} \]

    8. Simplified81.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(x, 0.5, -0.5\right)\right)} \]

    if -4.0000000000000003e-15 < (+.f64 x y)

    1. Initial program 100.0%

      \[\cos^{-1} \left(\frac{\left(\left(x + y\right) + z\right) - 1}{2}\right) \]
    2. Add Preprocessing

    3. Taylor expanded in x around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot \left(\left(y + z\right) - 1\right)\right)} \]
    4. Step-by-step derivation

      1. sub-negN/A

        \[\leadsto \cos^{-1} \left(\frac{1}{2} \cdot \color{blue}{\left(\left(y + z\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      2. metadata-evalN/A

        \[\leadsto \cos^{-1} \left(\frac{1}{2} \cdot \left(\left(y + z\right) + \color{blue}{-1}\right)\right) \]

      3. distribute-lft-inN/A

        \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot \left(y + z\right) + \frac{1}{2} \cdot -1\right)} \]

      4. metadata-evalN/A

        \[\leadsto \cos^{-1} \left(\frac{1}{2} \cdot \left(y + z\right) + \color{blue}{\frac{-1}{2}}\right) \]

      5. accelerator-lowering-fma.f64N/A

        \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, y + z, \frac{-1}{2}\right)\right)} \]

      6. +-lowering-+.f6498.8

        \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(0.5, \color{blue}{y + z}, -0.5\right)\right) \]

    5. Simplified98.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(0.5, y + z, -0.5\right)\right)} \]
  3. Recombined 2 regimes into one program.
  4. Add Preprocessing

Alternative 4: 97.2% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -4 \cdot 10^{-15}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(x, 0.5, -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(z, 0.5, -0.5\right)\right)\\ \end{array} \end{array} \]
(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -4e-15) (acos (fma x 0.5 -0.5)) (acos (fma z 0.5 -0.5))))
double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -4e-15) {
		tmp = acos(fma(x, 0.5, -0.5));
	} else {
		tmp = acos(fma(z, 0.5, -0.5));
	}
	return tmp;
}

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -4e-15)
		tmp = acos(fma(x, 0.5, -0.5));
	else
		tmp = acos(fma(z, 0.5, -0.5));
	end
	return tmp
end

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -4e-15], N[ArcCos[N[(x * 0.5 + -0.5), $MachinePrecision]], $MachinePrecision], N[ArcCos[N[(z * 0.5 + -0.5), $MachinePrecision]], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x + y \leq -4 \cdot 10^{-15}:\\
\;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(x, 0.5, -0.5\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(z, 0.5, -0.5\right)\right)\\


\end{array}
\end{array}
Derivation
  1. Split input into 2 regimes

  2. if (+.f64 x y) < -4.0000000000000003e-15

    1. Initial program 99.7%

      \[\cos^{-1} \left(\frac{\left(\left(x + y\right) + z\right) - 1}{2}\right) \]
    2. Add Preprocessing

    3. Taylor expanded in y around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot \left(\left(x + z\right) - 1\right)\right)} \]
    4. Step-by-step derivation

      1. sub-negN/A

        \[\leadsto \cos^{-1} \left(\frac{1}{2} \cdot \color{blue}{\left(\left(x + z\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      2. metadata-evalN/A

        \[\leadsto \cos^{-1} \left(\frac{1}{2} \cdot \left(\left(x + z\right) + \color{blue}{-1}\right)\right) \]

      3. distribute-lft-inN/A

        \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot \left(x + z\right) + \frac{1}{2} \cdot -1\right)} \]

      4. metadata-evalN/A

        \[\leadsto \cos^{-1} \left(\frac{1}{2} \cdot \left(x + z\right) + \color{blue}{\frac{-1}{2}}\right) \]

      5. accelerator-lowering-fma.f64N/A

        \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, x + z, \frac{-1}{2}\right)\right)} \]

      6. +-commutativeN/A

        \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{z + x}, \frac{-1}{2}\right)\right) \]

      7. +-lowering-+.f6483.0

        \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(0.5, \color{blue}{z + x}, -0.5\right)\right) \]

    5. Simplified83.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(0.5, z + x, -0.5\right)\right)} \]

    6. Taylor expanded in z around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot x - \frac{1}{2}\right)} \]
    7. Step-by-step derivation

      1. sub-negN/A

        \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

      2. *-commutativeN/A

        \[\leadsto \cos^{-1} \left(\color{blue}{x \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      3. metadata-evalN/A

        \[\leadsto \cos^{-1} \left(x \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}\right) \]

      4. accelerator-lowering-fma.f6481.8

        \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(x, 0.5, -0.5\right)\right)} \]

    8. Simplified81.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(x, 0.5, -0.5\right)\right)} \]

    if -4.0000000000000003e-15 < (+.f64 x y)

    1. Initial program 100.0%

      \[\cos^{-1} \left(\frac{\left(\left(x + y\right) + z\right) - 1}{2}\right) \]
    2. Add Preprocessing

    3. Taylor expanded in x around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot \left(\left(y + z\right) - 1\right)\right)} \]
    4. Step-by-step derivation

      1. sub-negN/A

        \[\leadsto \cos^{-1} \left(\frac{1}{2} \cdot \color{blue}{\left(\left(y + z\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      2. metadata-evalN/A

        \[\leadsto \cos^{-1} \left(\frac{1}{2} \cdot \left(\left(y + z\right) + \color{blue}{-1}\right)\right) \]

      3. distribute-lft-inN/A

        \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot \left(y + z\right) + \frac{1}{2} \cdot -1\right)} \]

      4. metadata-evalN/A

        \[\leadsto \cos^{-1} \left(\frac{1}{2} \cdot \left(y + z\right) + \color{blue}{\frac{-1}{2}}\right) \]

      5. accelerator-lowering-fma.f64N/A

        \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, y + z, \frac{-1}{2}\right)\right)} \]

      6. +-lowering-+.f6498.8

        \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(0.5, \color{blue}{y + z}, -0.5\right)\right) \]

    5. Simplified98.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(0.5, y + z, -0.5\right)\right)} \]

    6. Taylor expanded in y around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot z - \frac{1}{2}\right)} \]
    7. Step-by-step derivation

      1. sub-negN/A

        \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

      2. *-commutativeN/A

        \[\leadsto \cos^{-1} \left(\color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      3. metadata-evalN/A

        \[\leadsto \cos^{-1} \left(z \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}\right) \]

      4. accelerator-lowering-fma.f6498.4

        \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(z, 0.5, -0.5\right)\right)} \]

    8. Simplified98.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(z, 0.5, -0.5\right)\right)} \]
  3. Recombined 2 regimes into one program.
  4. Add Preprocessing

Alternative 5: 98.1% accurate, 1.1× speedup?

\[\begin{array}{l} \\ \cos^{-1} \left(\mathsf{fma}\left(0.5, x + z, -0.5\right)\right) \end{array} \]
(FPCore (x y z) :precision binary64 (acos (fma 0.5 (+ x z) -0.5)))
double code(double x, double y, double z) {
	return acos(fma(0.5, (x + z), -0.5));
}

function code(x, y, z)
	return acos(fma(0.5, Float64(x + z), -0.5))
end

code[x_, y_, z_] := N[ArcCos[N[(0.5 * N[(x + z), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\cos^{-1} \left(\mathsf{fma}\left(0.5, x + z, -0.5\right)\right)
\end{array}
Derivation

  1. Initial program 100.0%

    \[\cos^{-1} \left(\frac{\left(\left(x + y\right) + z\right) - 1}{2}\right) \]
  2. Add Preprocessing

  3. Taylor expanded in y around 0

    \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot \left(\left(x + z\right) - 1\right)\right)} \]
  4. Step-by-step derivation

    1. sub-negN/A

      \[\leadsto \cos^{-1} \left(\frac{1}{2} \cdot \color{blue}{\left(\left(x + z\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

    2. metadata-evalN/A

      \[\leadsto \cos^{-1} \left(\frac{1}{2} \cdot \left(\left(x + z\right) + \color{blue}{-1}\right)\right) \]

    3. distribute-lft-inN/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot \left(x + z\right) + \frac{1}{2} \cdot -1\right)} \]

    4. metadata-evalN/A

      \[\leadsto \cos^{-1} \left(\frac{1}{2} \cdot \left(x + z\right) + \color{blue}{\frac{-1}{2}}\right) \]

    5. accelerator-lowering-fma.f64N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, x + z, \frac{-1}{2}\right)\right)} \]

    6. +-commutativeN/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{z + x}, \frac{-1}{2}\right)\right) \]

    7. +-lowering-+.f6498.9

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(0.5, \color{blue}{z + x}, -0.5\right)\right) \]

  5. Simplified98.9%

    \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(0.5, z + x, -0.5\right)\right)} \]

  6. Final simplification98.9%

    \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(0.5, x + z, -0.5\right)\right) \]
  7. Add Preprocessing

Alternative 6: 96.2% accurate, 1.1× speedup?

\[\begin{array}{l} \\ \cos^{-1} \left(\mathsf{fma}\left(x, 0.5, -0.5\right)\right) \end{array} \]
(FPCore (x y z) :precision binary64 (acos (fma x 0.5 -0.5)))
double code(double x, double y, double z) {
	return acos(fma(x, 0.5, -0.5));
}

function code(x, y, z)
	return acos(fma(x, 0.5, -0.5))
end

code[x_, y_, z_] := N[ArcCos[N[(x * 0.5 + -0.5), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\cos^{-1} \left(\mathsf{fma}\left(x, 0.5, -0.5\right)\right)
\end{array}
Derivation

  1. Initial program 100.0%

    \[\cos^{-1} \left(\frac{\left(\left(x + y\right) + z\right) - 1}{2}\right) \]
  2. Add Preprocessing

  3. Taylor expanded in y around 0

    \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot \left(\left(x + z\right) - 1\right)\right)} \]
  4. Step-by-step derivation

    1. sub-negN/A

      \[\leadsto \cos^{-1} \left(\frac{1}{2} \cdot \color{blue}{\left(\left(x + z\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

    2. metadata-evalN/A

      \[\leadsto \cos^{-1} \left(\frac{1}{2} \cdot \left(\left(x + z\right) + \color{blue}{-1}\right)\right) \]

    3. distribute-lft-inN/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot \left(x + z\right) + \frac{1}{2} \cdot -1\right)} \]

    4. metadata-evalN/A

      \[\leadsto \cos^{-1} \left(\frac{1}{2} \cdot \left(x + z\right) + \color{blue}{\frac{-1}{2}}\right) \]

    5. accelerator-lowering-fma.f64N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, x + z, \frac{-1}{2}\right)\right)} \]

    6. +-commutativeN/A

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{z + x}, \frac{-1}{2}\right)\right) \]

    7. +-lowering-+.f6498.9

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(0.5, \color{blue}{z + x}, -0.5\right)\right) \]

  5. Simplified98.9%

    \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(0.5, z + x, -0.5\right)\right)} \]

  6. Taylor expanded in z around 0

    \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot x - \frac{1}{2}\right)} \]
  7. Step-by-step derivation

    1. sub-negN/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

    2. *-commutativeN/A

      \[\leadsto \cos^{-1} \left(\color{blue}{x \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

    3. metadata-evalN/A

      \[\leadsto \cos^{-1} \left(x \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}\right) \]

    4. accelerator-lowering-fma.f6497.4

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(x, 0.5, -0.5\right)\right)} \]

  8. Simplified97.4%

    \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(x, 0.5, -0.5\right)\right)} \]
  9. Add Preprocessing

Alternative 7: 20.4% accurate, 1.1× speedup?

\[\begin{array}{l} \\ \cos^{-1} \left(x \cdot 0.5\right) \end{array} \]
(FPCore (x y z) :precision binary64 (acos (* x 0.5)))
double code(double x, double y, double z) {
	return acos((x * 0.5));
}

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = acos((x * 0.5d0))
end function

public static double code(double x, double y, double z) {
	return Math.acos((x * 0.5));
}

def code(x, y, z):
	return math.acos((x * 0.5))

function code(x, y, z)
	return acos(Float64(x * 0.5))
end

function tmp = code(x, y, z)
	tmp = acos((x * 0.5));
end

code[x_, y_, z_] := N[ArcCos[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\cos^{-1} \left(x \cdot 0.5\right)
\end{array}
Derivation

  1. Initial program 100.0%

    \[\cos^{-1} \left(\frac{\left(\left(x + y\right) + z\right) - 1}{2}\right) \]
  2. Add Preprocessing

  3. Taylor expanded in x around inf

    \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{2} \cdot x\right)} \]
  4. Step-by-step derivation

    1. *-lowering-*.f6420.4

      \[\leadsto \cos^{-1} \color{blue}{\left(0.5 \cdot x\right)} \]

  5. Simplified20.4%

    \[\leadsto \cos^{-1} \color{blue}{\left(0.5 \cdot x\right)} \]

  6. Final simplification20.4%

    \[\leadsto \cos^{-1} \left(x \cdot 0.5\right) \]
  7. Add Preprocessing

Reproduce

?
herbie shell --seed 5 
(FPCore (x y z)
  :name "acos(((x + y + z) - 1)/2)"
  :precision binary64
  :pre (and (and (and (<= -1.0 x) (<= x 1.0)) (and (<= -1.0 y) (<= y 1.0))) (and (<= -1.0 z) (<= z 1.0)))
  (acos (/ (- (+ (+ x y) z) 1.0) 2.0)))