Ian Simplification

Percentage Accurate: 7.0% → 8.4%

Time: 1.2min

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 7.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 8.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\\ \frac{\log \left(e^{\mathsf{fma}\left({\pi}^{3}, 0.125, \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3}\right)\right) \cdot -8\right)}\right)}{\mathsf{fma}\left(\pi, \pi \cdot 0.25, t_0 \cdot \left(\pi + t_0 \cdot 4\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (sqrt (- 0.5 (* 0.5 x))))))
   (/
    (log
     (exp
      (fma
       (pow PI 3.0)
       0.125
       (*
        (expm1 (log1p (pow (- (* PI 0.5) (acos (sqrt (fma x -0.5 0.5)))) 3.0)))
        -8.0))))
    (fma PI (* PI 0.25) (* t_0 (+ PI (* t_0 4.0)))))))

C

double code(double x) {
	double t_0 = asin(sqrt((0.5 - (0.5 * x))));
	return log(exp(fma(pow(((double) M_PI), 3.0), 0.125, (expm1(log1p(pow(((((double) M_PI) * 0.5) - acos(sqrt(fma(x, -0.5, 0.5)))), 3.0))) * -8.0)))) / fma(((double) M_PI), (((double) M_PI) * 0.25), (t_0 * (((double) M_PI) + (t_0 * 4.0))));
}

Julia

function code(x)
	t_0 = asin(sqrt(Float64(0.5 - Float64(0.5 * x))))
	return Float64(log(exp(fma((pi ^ 3.0), 0.125, Float64(expm1(log1p((Float64(Float64(pi * 0.5) - acos(sqrt(fma(x, -0.5, 0.5)))) ^ 3.0))) * -8.0)))) / fma(pi, Float64(pi * 0.25), Float64(t_0 * Float64(pi + Float64(t_0 * 4.0)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[Log[N[Exp[N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.125 + N[(N[(Exp[N[Log[1 + N[Power[N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(Pi * N[(Pi * 0.25), $MachinePrecision] + N[(t$95$0 * N[(Pi + N[(t$95$0 * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.5%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

3. Applied egg-rr 7.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)}} \]
4. Step-by-step derivation

   1. asin-acos 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   2. div-inv 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   3. metadata-eval 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   4. cancel-sign-sub-inv 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   5. +-commutative 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{\left(-0.5\right) \cdot x + 0.5}}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   6. *-commutative 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{x \cdot \left(-0.5\right)} + 0.5}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   7. fma-def 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(x, -0.5, 0.5\right)}}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   8. metadata-eval 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \color{blue}{-0.5}, 0.5\right)}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

5. Applied egg-rr 9.3%

   \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]
6. Step-by-step derivation

   1. add-log-exp_binary64 9.3%

      \[\leadsto \color{blue}{\frac{\log \left(e^{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3} \cdot -8\right)}\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)}} \]

7. Applied rewrite-once 9.3%

   \[\leadsto \frac{\color{blue}{\log \left(e^{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3} \cdot -8\right)}\right)}}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]
8. Step-by-step derivation

   1. expm1-log1p-u_binary64 9.3%

      \[\leadsto \color{blue}{\frac{\log \left(e^{\mathsf{fma}\left({\pi}^{3}, 0.125, \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3}\right)\right) \cdot -8\right)}\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)}} \]

9. Applied rewrite-once 9.3%

   \[\leadsto \frac{\log \left(e^{\mathsf{fma}\left({\pi}^{3}, 0.125, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3}\right)\right)} \cdot -8\right)}\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

10. Final simplification 9.3%

    \[\leadsto \frac{\log \left(e^{\mathsf{fma}\left({\pi}^{3}, 0.125, \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3}\right)\right) \cdot -8\right)}\right)}{\mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\pi + \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4\right)\right)} \]

Alternative 2: 8.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\\ \frac{\log \left(e^{{\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3} \cdot -8 + {\pi}^{3} \cdot 0.125}\right)}{\mathsf{fma}\left(\pi, \pi \cdot 0.25, t_0 \cdot \left(\pi + t_0 \cdot 4\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (sqrt (- 0.5 (* 0.5 x))))))
   (/
    (log
     (exp
      (+
       (* (pow (- (* PI 0.5) (acos (sqrt (fma x -0.5 0.5)))) 3.0) -8.0)
       (* (pow PI 3.0) 0.125))))
    (fma PI (* PI 0.25) (* t_0 (+ PI (* t_0 4.0)))))))

C

double code(double x) {
	double t_0 = asin(sqrt((0.5 - (0.5 * x))));
	return log(exp(((pow(((((double) M_PI) * 0.5) - acos(sqrt(fma(x, -0.5, 0.5)))), 3.0) * -8.0) + (pow(((double) M_PI), 3.0) * 0.125)))) / fma(((double) M_PI), (((double) M_PI) * 0.25), (t_0 * (((double) M_PI) + (t_0 * 4.0))));
}

Julia

function code(x)
	t_0 = asin(sqrt(Float64(0.5 - Float64(0.5 * x))))
	return Float64(log(exp(Float64(Float64((Float64(Float64(pi * 0.5) - acos(sqrt(fma(x, -0.5, 0.5)))) ^ 3.0) * -8.0) + Float64((pi ^ 3.0) * 0.125)))) / fma(pi, Float64(pi * 0.25), Float64(t_0 * Float64(pi + Float64(t_0 * 4.0)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[Log[N[Exp[N[(N[(N[Power[N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * -8.0), $MachinePrecision] + N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(Pi * N[(Pi * 0.25), $MachinePrecision] + N[(t$95$0 * N[(Pi + N[(t$95$0 * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.5%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

3. Applied egg-rr 7.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)}} \]
4. Step-by-step derivation

   1. asin-acos 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   2. div-inv 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   3. metadata-eval 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   4. cancel-sign-sub-inv 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   5. +-commutative 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{\left(-0.5\right) \cdot x + 0.5}}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   6. *-commutative 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{x \cdot \left(-0.5\right)} + 0.5}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   7. fma-def 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(x, -0.5, 0.5\right)}}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   8. metadata-eval 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \color{blue}{-0.5}, 0.5\right)}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

5. Applied egg-rr 9.3%

   \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]
6. Step-by-step derivation

   1. add-log-exp_binary64 9.3%

      \[\leadsto \color{blue}{\frac{\log \left(e^{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3} \cdot -8\right)}\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)}} \]

7. Applied rewrite-once 9.3%

   \[\leadsto \frac{\color{blue}{\log \left(e^{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3} \cdot -8\right)}\right)}}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

8. Taylor expanded in x around 0 9.3%

   \[\leadsto \frac{\log \left(e^{\color{blue}{-8 \cdot {\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3} + 0.125 \cdot {\pi}^{3}}}\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

9. Final simplification 9.3%

   \[\leadsto \frac{\log \left(e^{{\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3} \cdot -8 + {\pi}^{3} \cdot 0.125}\right)}{\mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\pi + \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4\right)\right)} \]

Alternative 3: 8.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\\ \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, \pi \cdot 0.25, t_0 \cdot \left(\pi + t_0 \cdot 4\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (sqrt (- 0.5 (* 0.5 x))))))
   (/
    (fma
     (pow PI 3.0)
     0.125
     (* (pow (- (* PI 0.5) (acos (sqrt (fma x -0.5 0.5)))) 3.0) -8.0))
    (fma PI (* PI 0.25) (* t_0 (+ PI (* t_0 4.0)))))))

C

double code(double x) {
	double t_0 = asin(sqrt((0.5 - (0.5 * x))));
	return fma(pow(((double) M_PI), 3.0), 0.125, (pow(((((double) M_PI) * 0.5) - acos(sqrt(fma(x, -0.5, 0.5)))), 3.0) * -8.0)) / fma(((double) M_PI), (((double) M_PI) * 0.25), (t_0 * (((double) M_PI) + (t_0 * 4.0))));
}

Julia

function code(x)
	t_0 = asin(sqrt(Float64(0.5 - Float64(0.5 * x))))
	return Float64(fma((pi ^ 3.0), 0.125, Float64((Float64(Float64(pi * 0.5) - acos(sqrt(fma(x, -0.5, 0.5)))) ^ 3.0) * -8.0)) / fma(pi, Float64(pi * 0.25), Float64(t_0 * Float64(pi + Float64(t_0 * 4.0)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.125 + N[(N[Power[N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision] / N[(Pi * N[(Pi * 0.25), $MachinePrecision] + N[(t$95$0 * N[(Pi + N[(t$95$0 * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.5%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

3. Applied egg-rr 7.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)}} \]
4. Step-by-step derivation

   1. asin-acos 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   2. div-inv 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   3. metadata-eval 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   4. cancel-sign-sub-inv 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   5. +-commutative 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{\left(-0.5\right) \cdot x + 0.5}}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   6. *-commutative 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{x \cdot \left(-0.5\right)} + 0.5}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   7. fma-def 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(x, -0.5, 0.5\right)}}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   8. metadata-eval 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \color{blue}{-0.5}, 0.5\right)}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

5. Applied egg-rr 9.3%

   \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

6. Final simplification 9.3%

   \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, \pi \cdot 0.25, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\pi + \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4\right)\right)} \]

Alternative 4: 8.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot 2\right)}^{2} - {\left(\pi \cdot 0.5\right)}^{2}}{\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2 - \pi \cdot 0.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (-
   (pow (* (- (* PI 0.5) (acos (sqrt (fma x -0.5 0.5)))) 2.0) 2.0)
   (pow (* PI 0.5) 2.0))
  (- (* (asin (sqrt (- 0.5 (* 0.5 x)))) -2.0) (* PI 0.5))))

C

double code(double x) {
	return (pow((((((double) M_PI) * 0.5) - acos(sqrt(fma(x, -0.5, 0.5)))) * 2.0), 2.0) - pow((((double) M_PI) * 0.5), 2.0)) / ((asin(sqrt((0.5 - (0.5 * x)))) * -2.0) - (((double) M_PI) * 0.5));
}

Julia

function code(x)
	return Float64(Float64((Float64(Float64(Float64(pi * 0.5) - acos(sqrt(fma(x, -0.5, 0.5)))) * 2.0) ^ 2.0) - (Float64(pi * 0.5) ^ 2.0)) / Float64(Float64(asin(sqrt(Float64(0.5 - Float64(0.5 * x)))) * -2.0) - Float64(pi * 0.5)))
end

Wolfram

code[x_] := N[(N[(N[Power[N[(N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[Sqrt[N[(x * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(Pi * 0.5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[ArcSin[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] - N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.5%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Step-by-step derivation

   1. sub-neg 7.5%

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

   2. +-commutative 7.5%

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

   3. flip-+ 7.5%

      \[\leadsto \color{blue}{\frac{\left(-2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) \cdot \left(-2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) - \frac{\pi}{2} \cdot \frac{\pi}{2}}{\left(-2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right) - \frac{\pi}{2}}} \]

3. Applied egg-rr 7.5%

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}^{2} - {\left(0.5 \cdot \pi\right)}^{2}}{-2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) - 0.5 \cdot \pi}} \]
4. Step-by-step derivation

   1. asin-acos 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   2. div-inv 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\color{blue}{\pi \cdot \frac{1}{2}} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   3. metadata-eval 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot \color{blue}{0.5} - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   4. cancel-sign-sub-inv 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{0.5 + \left(-0.5\right) \cdot x}}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   5. +-commutative 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{\left(-0.5\right) \cdot x + 0.5}}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   6. *-commutative 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{x \cdot \left(-0.5\right)} + 0.5}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   7. fma-def 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(x, -0.5, 0.5\right)}}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

   8. metadata-eval 9.3%

      \[\leadsto \frac{\mathsf{fma}\left({\pi}^{3}, 0.125, {\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \color{blue}{-0.5}, 0.5\right)}\right)\right)}^{3} \cdot -8\right)}{\mathsf{fma}\left(\pi, 0.25 \cdot \pi, \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot \left(\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot 4 + \pi\right)\right)} \]

5. Applied egg-rr 9.3%

   \[\leadsto \frac{{\left(2 \cdot \color{blue}{\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)}\right)}^{2} - {\left(0.5 \cdot \pi\right)}^{2}}{-2 \cdot \sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) - 0.5 \cdot \pi} \]

6. Final simplification 9.3%

   \[\leadsto \frac{{\left(\left(\pi \cdot 0.5 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right) \cdot 2\right)}^{2} - {\left(\pi \cdot 0.5\right)}^{2}}{\sin^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) \cdot -2 - \pi \cdot 0.5} \]

Alternative 5: 8.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \log \left(e^{\frac{\pi}{2} + 2 \cdot \left(\cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) - \pi \cdot 0.5\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log
  (exp (+ (/ PI 2.0) (* 2.0 (- (acos (sqrt (- 0.5 (* 0.5 x)))) (* PI 0.5)))))))

C

double code(double x) {
	return log(exp(((((double) M_PI) / 2.0) + (2.0 * (acos(sqrt((0.5 - (0.5 * x)))) - (((double) M_PI) * 0.5))))));
}

Java

public static double code(double x) {
	return Math.log(Math.exp(((Math.PI / 2.0) + (2.0 * (Math.acos(Math.sqrt((0.5 - (0.5 * x)))) - (Math.PI * 0.5))))));
}

Python

def code(x):
	return math.log(math.exp(((math.pi / 2.0) + (2.0 * (math.acos(math.sqrt((0.5 - (0.5 * x)))) - (math.pi * 0.5))))))

Julia

function code(x)
	return log(exp(Float64(Float64(pi / 2.0) + Float64(2.0 * Float64(acos(sqrt(Float64(0.5 - Float64(0.5 * x)))) - Float64(pi * 0.5))))))
end

MATLAB

function tmp = code(x)
	tmp = log(exp(((pi / 2.0) + (2.0 * (acos(sqrt((0.5 - (0.5 * x)))) - (pi * 0.5))))));
end

Wolfram

code[x_] := N[Log[N[Exp[N[(N[(Pi / 2.0), $MachinePrecision] + N[(2.0 * N[(N[ArcCos[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] - N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 7.5%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Step-by-step derivation

   1. asin-acos 9.3%

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

   2. sub-neg 9.3%

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

   3. div-inv 9.3%

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

   4. metadata-eval 9.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\pi \cdot \color{blue}{0.5} + \left(-\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \]

   5. *-commutative 9.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\color{blue}{0.5 \cdot \pi} + \left(-\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \]

   6. div-sub 9.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(0.5 \cdot \pi + \left(-\cos^{-1} \left(\sqrt{\color{blue}{\frac{1}{2} - \frac{x}{2}}}\right)\right)\right) \]

   7. metadata-eval 9.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(0.5 \cdot \pi + \left(-\cos^{-1} \left(\sqrt{\color{blue}{0.5} - \frac{x}{2}}\right)\right)\right) \]

   8. div-inv 9.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(0.5 \cdot \pi + \left(-\cos^{-1} \left(\sqrt{0.5 - \color{blue}{x \cdot \frac{1}{2}}}\right)\right)\right) \]

   9. metadata-eval 9.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(0.5 \cdot \pi + \left(-\cos^{-1} \left(\sqrt{0.5 - x \cdot \color{blue}{0.5}}\right)\right)\right) \]

   10. *-commutative 9.3%

       \[\leadsto \frac{\pi}{2} - 2 \cdot \left(0.5 \cdot \pi + \left(-\cos^{-1} \left(\sqrt{0.5 - \color{blue}{0.5 \cdot x}}\right)\right)\right) \]

3. Applied egg-rr 9.3%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left(0.5 \cdot \pi + \left(-\cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)} \]
4. Step-by-step derivation

   1. sub-neg 9.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

5. Simplified 9.3%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]
6. Step-by-step derivation

   1. add-log-exp_binary64 9.3%

      \[\leadsto \color{blue}{\log \left(e^{\frac{\pi}{2} - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}\right)} \]

7. Applied rewrite-once 9.3%

   \[\leadsto \color{blue}{\log \left(e^{\frac{\pi}{2} - 2 \cdot \left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)}\right)} \]

8. Final simplification 9.3%

   \[\leadsto \log \left(e^{\frac{\pi}{2} + 2 \cdot \left(\cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) - \pi \cdot 0.5\right)}\right) \]

Alternative 6: 8.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} + 2 \cdot \left(\cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) - \pi \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (/ PI 2.0) (* 2.0 (- (acos (sqrt (- 0.5 (* 0.5 x)))) (* PI 0.5)))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) + (2.0 * (acos(sqrt((0.5 - (0.5 * x)))) - (((double) M_PI) * 0.5)));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) + (2.0 * (Math.acos(Math.sqrt((0.5 - (0.5 * x)))) - (Math.PI * 0.5)));
}

Python

def code(x):
	return (math.pi / 2.0) + (2.0 * (math.acos(math.sqrt((0.5 - (0.5 * x)))) - (math.pi * 0.5)))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) + Float64(2.0 * Float64(acos(sqrt(Float64(0.5 - Float64(0.5 * x)))) - Float64(pi * 0.5))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) + (2.0 * (acos(sqrt((0.5 - (0.5 * x)))) - (pi * 0.5)));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] + N[(2.0 * N[(N[ArcCos[N[Sqrt[N[(0.5 - N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] - N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.5%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Step-by-step derivation

   1. asin-acos 9.3%

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

   2. sub-neg 9.3%

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

   3. div-inv 9.3%

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

   4. metadata-eval 9.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\pi \cdot \color{blue}{0.5} + \left(-\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \]

   5. *-commutative 9.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(\color{blue}{0.5 \cdot \pi} + \left(-\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\right)\right) \]

   6. div-sub 9.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(0.5 \cdot \pi + \left(-\cos^{-1} \left(\sqrt{\color{blue}{\frac{1}{2} - \frac{x}{2}}}\right)\right)\right) \]

   7. metadata-eval 9.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(0.5 \cdot \pi + \left(-\cos^{-1} \left(\sqrt{\color{blue}{0.5} - \frac{x}{2}}\right)\right)\right) \]

   8. div-inv 9.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(0.5 \cdot \pi + \left(-\cos^{-1} \left(\sqrt{0.5 - \color{blue}{x \cdot \frac{1}{2}}}\right)\right)\right) \]

   9. metadata-eval 9.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \left(0.5 \cdot \pi + \left(-\cos^{-1} \left(\sqrt{0.5 - x \cdot \color{blue}{0.5}}\right)\right)\right) \]

   10. *-commutative 9.3%

       \[\leadsto \frac{\pi}{2} - 2 \cdot \left(0.5 \cdot \pi + \left(-\cos^{-1} \left(\sqrt{0.5 - \color{blue}{0.5 \cdot x}}\right)\right)\right) \]

3. Applied egg-rr 9.3%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left(0.5 \cdot \pi + \left(-\cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)\right)} \]
4. Step-by-step derivation

   1. sub-neg 9.3%

      \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

5. Simplified 9.3%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \color{blue}{\left(0.5 \cdot \pi - \cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right)\right)} \]

6. Final simplification 9.3%

   \[\leadsto \frac{\pi}{2} + 2 \cdot \left(\cos^{-1} \left(\sqrt{0.5 - 0.5 \cdot x}\right) - \pi \cdot 0.5\right) \]

Alternative 7: 7.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.5%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

2. Final simplification 7.5%

   \[\leadsto \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

Alternative 8: 4.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \langle \left( \frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{0.5 - \frac{x}{2}}\right) \right)_{\text{binary32}} \rangle_{\text{binary64}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (cast
  (!
   :precision
   binary32
   (+ (/ PI 2.0) (* -2.0 (asin (sqrt (- 0.5 (/ x 2.0)))))))))

C

double code(double x) {
	float tmp = (((float) M_PI) / 2.0f) + (-2.0f * asinf(sqrtf((0.5f - (x / 2.0f)))));
	return (double) tmp;
}

Julia

function code(x)
	tmp = Float32(Float32(Float32(pi) / Float32(2.0)) + Float32(Float32(-2.0) * asin(sqrt(Float32(Float32(0.5) - Float32(x / Float32(2.0)))))))
	return Float64(tmp)
end

MATLAB

function tmp_2 = code(x)
	tmp = (single(pi) / single(2.0)) + (single(-2.0) * asin(sqrt((single(0.5) - single((x / double(single(2.0))))))));
	tmp_2 = double(tmp);
end

Derivation

1. Initial program 5.3%

   \[\langle \left( \frac{\pi}{2} + \sin^{-1} \left(\sqrt{0.5 - \frac{x}{2}}\right) \cdot -2 \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

2. Final simplification 5.3%

   \[\leadsto \langle \left( \frac{\pi}{2} + -2 \cdot \sin^{-1} \left(\sqrt{0.5 - \frac{x}{2}}\right) \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

Developer target: 100.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (asin x))

C

double code(double x) {
	return asin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = asin(x)
end function

Java

public static double code(double x) {
	return Math.asin(x);
}

Python

def code(x):
	return math.asin(x)

Julia

function code(x)
	return asin(x)
end

MATLAB

function tmp = code(x)
	tmp = asin(x);
end

Wolfram

code[x_] := N[ArcSin[x], $MachinePrecision]

Reproduce

herbie shell --seed 2023297 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :herbie-target
  (asin x)

  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))