bug323 (missed optimization)

Percentage Accurate: 6.7% → 10.2%

Time: 9.6s

Alternatives: 14

Speedup: 1.0×

Specification

\[0 \leq x \land x \leq 0.5\]

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Initial Program: 6.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Alternative 1: 10.2% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (- (* PI 0.5) (fma (* 0.5 (sqrt PI)) (sqrt PI) (- (acos (- 1.0 x))))))

C

double code(double x) {
	return (((double) M_PI) * 0.5) - fma((0.5 * sqrt(((double) M_PI))), sqrt(((double) M_PI)), -acos((1.0 - x)));
}

Julia

function code(x)
	return Float64(Float64(pi * 0.5) - fma(Float64(0.5 * sqrt(pi)), sqrt(pi), Float64(-acos(Float64(1.0 - x)))))
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[(N[(0.5 * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision] + (-N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin 7.3%

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

   2. sub-neg 7.3%

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

   3. div-inv 7.3%

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

   4. metadata-eval 7.3%

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

3. Applied egg-rr 7.3%

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

   1. sub-neg 7.3%

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

5. Simplified 7.3%

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

   1. asin-acos 7.3%

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

   2. div-inv 7.3%

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

   3. metadata-eval 7.3%

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

   4. *-commutative 7.3%

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

   5. add-sqr-sqrt 10.6%

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

   6. associate-*r* 10.6%

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

   7. fma-neg 10.6%

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

7. Applied egg-rr 10.6%

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

8. Final simplification 10.6%

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

Alternative 2: 10.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sin^{-1} \left(1 - x\right)}\\ \pi \cdot 0.5 - t_0 \cdot {t_0}^{2} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (asin (- 1.0 x))))) (- (* PI 0.5) (* t_0 (pow t_0 2.0)))))

C

double code(double x) {
	double t_0 = cbrt(asin((1.0 - x)));
	return (((double) M_PI) * 0.5) - (t_0 * pow(t_0, 2.0));
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt(Math.asin((1.0 - x)));
	return (Math.PI * 0.5) - (t_0 * Math.pow(t_0, 2.0));
}

Julia

function code(x)
	t_0 = cbrt(asin(Float64(1.0 - x)))
	return Float64(Float64(pi * 0.5) - Float64(t_0 * (t_0 ^ 2.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(Pi * 0.5), $MachinePrecision] - N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin 7.3%

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

   2. sub-neg 7.3%

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

   3. div-inv 7.3%

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

   4. metadata-eval 7.3%

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

3. Applied egg-rr 7.3%

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

   1. sub-neg 7.3%

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

5. Simplified 7.3%

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

   1. expm1-log1p-u 7.3%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)} \]

7. Applied egg-rr 7.3%

   \[\leadsto \pi \cdot 0.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)} \]
8. Step-by-step derivation

   1. expm1-log1p-u 7.3%

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

   2. add-cube-cbrt 10.6%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

   3. pow2 10.6%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)} \]

9. Applied egg-rr 10.6%

   \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

10. Final simplification 10.6%

    \[\leadsto \pi \cdot 0.5 - \sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{2} \]

Alternative 3: 10.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \log \left(e \cdot {\left({\left(e^{\cos^{-1} \left(1 - x\right) + -1}\right)}^{3}\right)}^{0.3333333333333333}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log
  (* E (pow (pow (exp (+ (acos (- 1.0 x)) -1.0)) 3.0) 0.3333333333333333))))

C

double code(double x) {
	return log((((double) M_E) * pow(pow(exp((acos((1.0 - x)) + -1.0)), 3.0), 0.3333333333333333)));
}

Java

public static double code(double x) {
	return Math.log((Math.E * Math.pow(Math.pow(Math.exp((Math.acos((1.0 - x)) + -1.0)), 3.0), 0.3333333333333333)));
}

Python

def code(x):
	return math.log((math.e * math.pow(math.pow(math.exp((math.acos((1.0 - x)) + -1.0)), 3.0), 0.3333333333333333)))

Julia

function code(x)
	return log(Float64(exp(1) * ((exp(Float64(acos(Float64(1.0 - x)) + -1.0)) ^ 3.0) ^ 0.3333333333333333)))
end

MATLAB

function tmp = code(x)
	tmp = log((2.71828182845904523536 * ((exp((acos((1.0 - x)) + -1.0)) ^ 3.0) ^ 0.3333333333333333)));
end

Wolfram

code[x_] := N[Log[N[(E * N[Power[N[Power[N[Exp[N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. add-log-exp 7.3%

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

3. Applied egg-rr 7.3%

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

   1. expm1-log1p-u 7.3%

      \[\leadsto \log \left(e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)}}\right) \]

   2. expm1-udef 7.3%

      \[\leadsto \log \left(e^{\color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1}}\right) \]

   3. log1p-udef 7.3%

      \[\leadsto \log \left(e^{e^{\color{blue}{\log \left(1 + \cos^{-1} \left(1 - x\right)\right)}} - 1}\right) \]

   4. add-exp-log 7.3%

      \[\leadsto \log \left(e^{\color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1}\right) \]

   5. associate--l+ 7.3%

      \[\leadsto \log \left(e^{\color{blue}{1 + \left(\cos^{-1} \left(1 - x\right) - 1\right)}}\right) \]

   6. exp-sum 7.3%

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

   7. exp-1-e 7.3%

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

   8. sub-neg 7.3%

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

   9. metadata-eval 7.3%

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

5. Applied egg-rr 7.3%

   \[\leadsto \log \color{blue}{\left(e \cdot e^{\cos^{-1} \left(1 - x\right) + -1}\right)} \]
6. Step-by-step derivation

   1. add-cbrt-cube 7.3%

      \[\leadsto \log \left(e \cdot \color{blue}{\sqrt[3]{\left(e^{\cos^{-1} \left(1 - x\right) + -1} \cdot e^{\cos^{-1} \left(1 - x\right) + -1}\right) \cdot e^{\cos^{-1} \left(1 - x\right) + -1}}}\right) \]

   2. pow1/3 10.6%

      \[\leadsto \log \left(e \cdot \color{blue}{{\left(\left(e^{\cos^{-1} \left(1 - x\right) + -1} \cdot e^{\cos^{-1} \left(1 - x\right) + -1}\right) \cdot e^{\cos^{-1} \left(1 - x\right) + -1}\right)}^{0.3333333333333333}}\right) \]

   3. pow3 10.6%

      \[\leadsto \log \left(e \cdot {\color{blue}{\left({\left(e^{\cos^{-1} \left(1 - x\right) + -1}\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

7. Applied egg-rr 10.6%

   \[\leadsto \log \left(e \cdot \color{blue}{{\left({\left(e^{\cos^{-1} \left(1 - x\right) + -1}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

8. Final simplification 10.6%

   \[\leadsto \log \left(e \cdot {\left({\left(e^{\cos^{-1} \left(1 - x\right) + -1}\right)}^{3}\right)}^{0.3333333333333333}\right) \]

Alternative 4: 10.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \pi \cdot 0.5 - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* PI 0.5) (expm1 (log1p (pow (cbrt (asin (- 1.0 x))) 3.0)))))

C

double code(double x) {
	return (((double) M_PI) * 0.5) - expm1(log1p(pow(cbrt(asin((1.0 - x))), 3.0)));
}

Java

public static double code(double x) {
	return (Math.PI * 0.5) - Math.expm1(Math.log1p(Math.pow(Math.cbrt(Math.asin((1.0 - x))), 3.0)));
}

Julia

function code(x)
	return Float64(Float64(pi * 0.5) - expm1(log1p((cbrt(asin(Float64(1.0 - x))) ^ 3.0))))
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[(Exp[N[Log[1 + N[Power[N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin 7.3%

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

   2. sub-neg 7.3%

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

   3. div-inv 7.3%

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

   4. metadata-eval 7.3%

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

3. Applied egg-rr 7.3%

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

   1. sub-neg 7.3%

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

5. Simplified 7.3%

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

   1. expm1-log1p-u 7.3%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)} \]

7. Applied egg-rr 7.3%

   \[\leadsto \pi \cdot 0.5 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(1 - x\right)\right)\right)} \]
8. Step-by-step derivation

   1. add-cube-cbrt 10.6%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

   2. pow3 10.6%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

9. Applied egg-rr 10.6%

   \[\leadsto \pi \cdot 0.5 - \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}}\right)\right) \]

10. Final simplification 10.6%

    \[\leadsto \pi \cdot 0.5 - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}\right)\right) \]

Alternative 5: 9.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\pi - t_0\\ \mathbf{else}:\\ \;\;\;\;-1 + e^{\mathsf{log1p}\left(1 + \left(t_0 + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0) (- PI t_0) (+ -1.0 (exp (log1p (+ 1.0 (+ t_0 -1.0))))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((double) M_PI) - t_0;
	} else {
		tmp = -1.0 + exp(log1p((1.0 + (t_0 + -1.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.PI - t_0;
	} else {
		tmp = -1.0 + Math.exp(Math.log1p((1.0 + (t_0 + -1.0))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.pi - t_0
	else:
		tmp = -1.0 + math.exp(math.log1p((1.0 + (t_0 + -1.0))))
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(pi - t_0);
	else
		tmp = Float64(-1.0 + exp(log1p(Float64(1.0 + Float64(t_0 + -1.0)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(Pi - t$95$0), $MachinePrecision], N[(-1.0 + N[Exp[N[Log[1 + N[(1.0 + N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 1 x)) < 0.0

   1. Initial program 3.9%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   5. Simplified 3.9%

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

   6. Taylor expanded in x around 0 3.9%

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

      1. fma-neg 3.9%

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

      2. rem-square-sqrt 0.0%

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

      3. fabs-sqr 0.0%

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

      4. rem-square-sqrt 6.5%

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

      5. fabs-neg 6.5%

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

      6. rem-square-sqrt 6.5%

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

      7. fabs-sqr 6.5%

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

      8. rem-square-sqrt 6.5%

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

   8. Simplified 6.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \pi, \sin^{-1} \left(1 - x\right)\right)} \]
   9. Step-by-step derivation

      1. fma-udef 6.5%

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

      2. *-commutative 6.5%

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

      3. asin-acos 6.5%

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

      4. div-inv 6.5%

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

      5. metadata-eval 6.5%

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

      6. associate-+r- 6.5%

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

   10. Applied egg-rr 6.5%

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

       1. distribute-lft-out 6.5%

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

       2. metadata-eval 6.5%

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

       3. *-rgt-identity 6.5%

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

   12. Simplified 6.5%

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

   if 0.0 < (acos.f64 (-.f64 1 x))

   1. Initial program 70.9%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. add-cube-cbrt 71.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\cos^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\cos^{-1} \left(1 - x\right)}} \]

      2. pow3 71.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{3}} \]

   3. Applied egg-rr 71.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{3}} \]
   4. Step-by-step derivation

      1. rem-cube-cbrt 70.9%

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

      2. expm1-log1p-u 70.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      3. expm1-udef 71.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      4. log1p-udef 71.0%

         \[\leadsto e^{\color{blue}{\log \left(1 + \cos^{-1} \left(1 - x\right)\right)}} - 1 \]

      5. *-rgt-identity 71.0%

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

      6. add-exp-log 71.0%

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

      7. *-rgt-identity 71.0%

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

      8. associate--l+ 70.9%

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

      9. +-commutative 70.9%

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

      10. rem-cube-cbrt 70.9%

          \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{3}} - 1\right) + 1 \]

      11. sqr-pow 70.9%

          \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{\left(\frac{3}{2}\right)}} - 1\right) + 1 \]

      12. difference-of-sqr-1 71.2%

          \[\leadsto \color{blue}{\left({\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{\left(\frac{3}{2}\right)} + 1\right) \cdot \left({\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{\left(\frac{3}{2}\right)} - 1\right)} + 1 \]

      13. fma-def 71.2%

          \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{\left(\frac{3}{2}\right)} + 1, {\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{\left(\frac{3}{2}\right)} - 1, 1\right)} \]

   5. Applied egg-rr 70.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\cos^{-1} \left(1 - x\right)} + 1, \sqrt{\cos^{-1} \left(1 - x\right)} - 1, 1\right)} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 70.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\sqrt{\cos^{-1} \left(1 - x\right)} + 1, \sqrt{\cos^{-1} \left(1 - x\right)} - 1, 1\right)\right)\right)} \]

      2. expm1-udef 70.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\sqrt{\cos^{-1} \left(1 - x\right)} + 1, \sqrt{\cos^{-1} \left(1 - x\right)} - 1, 1\right)\right)} - 1} \]

      3. fma-udef 70.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\cos^{-1} \left(1 - x\right)} + 1\right) \cdot \left(\sqrt{\cos^{-1} \left(1 - x\right)} - 1\right) + 1}\right)} - 1 \]

      4. difference-of-sqr-1 71.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\cos^{-1} \left(1 - x\right)} \cdot \sqrt{\cos^{-1} \left(1 - x\right)} - 1\right)} + 1\right)} - 1 \]

      5. add-sqr-sqrt 71.2%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\color{blue}{\cos^{-1} \left(1 - x\right)} - 1\right) + 1\right)} - 1 \]

      6. sub-neg 71.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\cos^{-1} \left(1 - x\right) + \left(-1\right)\right)} + 1\right)} - 1 \]

      7. metadata-eval 71.2%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\cos^{-1} \left(1 - x\right) + \color{blue}{-1}\right) + 1\right)} - 1 \]

   7. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1\right)} - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + e^{\mathsf{log1p}\left(1 + \left(\cos^{-1} \left(1 - x\right) + -1\right)\right)}\\ \end{array} \]

Alternative 6: 9.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\pi - t_0\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{-1 + \left(1 + t_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0) (- PI t_0) (log (exp (+ -1.0 (+ 1.0 t_0)))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((double) M_PI) - t_0;
	} else {
		tmp = log(exp((-1.0 + (1.0 + t_0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.PI - t_0;
	} else {
		tmp = Math.log(Math.exp((-1.0 + (1.0 + t_0))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.pi - t_0
	else:
		tmp = math.log(math.exp((-1.0 + (1.0 + t_0))))
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(pi - t_0);
	else
		tmp = log(exp(Float64(-1.0 + Float64(1.0 + t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = pi - t_0;
	else
		tmp = log(exp((-1.0 + (1.0 + t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(Pi - t$95$0), $MachinePrecision], N[Log[N[Exp[N[(-1.0 + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 1 x)) < 0.0

   1. Initial program 3.9%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 3.9%

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

      2. sub-neg 3.9%

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

      3. div-inv 3.9%

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

      4. metadata-eval 3.9%

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

   3. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

   5. Simplified 3.9%

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

   6. Taylor expanded in x around 0 3.9%

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

      1. fma-neg 3.9%

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

      2. rem-square-sqrt 0.0%

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

      3. fabs-sqr 0.0%

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

      4. rem-square-sqrt 6.5%

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

      5. fabs-neg 6.5%

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

      6. rem-square-sqrt 6.5%

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

      7. fabs-sqr 6.5%

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

      8. rem-square-sqrt 6.5%

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

   8. Simplified 6.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \pi, \sin^{-1} \left(1 - x\right)\right)} \]
   9. Step-by-step derivation

      1. fma-udef 6.5%

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

      2. *-commutative 6.5%

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

      3. asin-acos 6.5%

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

      4. div-inv 6.5%

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

      5. metadata-eval 6.5%

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

      6. associate-+r- 6.5%

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

   10. Applied egg-rr 6.5%

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

       1. distribute-lft-out 6.5%

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

       2. metadata-eval 6.5%

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

       3. *-rgt-identity 6.5%

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

   12. Simplified 6.5%

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

   if 0.0 < (acos.f64 (-.f64 1 x))

   1. Initial program 70.9%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. add-log-exp 71.0%

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

   3. Applied egg-rr 71.0%

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

      1. expm1-log1p-u 70.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      2. expm1-udef 71.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      3. log1p-udef 71.0%

         \[\leadsto e^{\color{blue}{\log \left(1 + \cos^{-1} \left(1 - x\right)\right)}} - 1 \]

      4. add-exp-log 71.0%

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

   5. Applied egg-rr 71.2%

      \[\leadsto \log \left(e^{\color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right) - 1}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{-1 + \left(1 + \cos^{-1} \left(1 - x\right)\right)}\right)\\ \end{array} \]

Alternative 7: 10.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \pi \cdot 0.5 - {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* PI 0.5) (pow (cbrt (asin (- 1.0 x))) 3.0)))

C

double code(double x) {
	return (((double) M_PI) * 0.5) - pow(cbrt(asin((1.0 - x))), 3.0);
}

Java

public static double code(double x) {
	return (Math.PI * 0.5) - Math.pow(Math.cbrt(Math.asin((1.0 - x))), 3.0);
}

Julia

function code(x)
	return Float64(Float64(pi * 0.5) - (cbrt(asin(Float64(1.0 - x))) ^ 3.0))
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin 7.3%

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

   2. sub-neg 7.3%

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

   3. div-inv 7.3%

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

   4. metadata-eval 7.3%

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

3. Applied egg-rr 7.3%

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

   1. sub-neg 7.3%

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

5. Simplified 7.3%

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

   1. add-cube-cbrt 10.6%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

   2. pow3 10.6%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

7. Applied egg-rr 10.6%

   \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

8. Final simplification 10.6%

   \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3} \]

Alternative 8: 10.2% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (- (* PI 0.5) (pow (sqrt (asin (- 1.0 x))) 2.0)))

C

double code(double x) {
	return (((double) M_PI) * 0.5) - pow(sqrt(asin((1.0 - x))), 2.0);
}

Java

public static double code(double x) {
	return (Math.PI * 0.5) - Math.pow(Math.sqrt(Math.asin((1.0 - x))), 2.0);
}

Python

def code(x):
	return (math.pi * 0.5) - math.pow(math.sqrt(math.asin((1.0 - x))), 2.0)

Julia

function code(x)
	return Float64(Float64(pi * 0.5) - (sqrt(asin(Float64(1.0 - x))) ^ 2.0))
end

MATLAB

function tmp = code(x)
	tmp = (pi * 0.5) - (sqrt(asin((1.0 - x))) ^ 2.0);
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[Power[N[Sqrt[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin 7.3%

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

   2. sub-neg 7.3%

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

   3. div-inv 7.3%

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

   4. metadata-eval 7.3%

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

3. Applied egg-rr 7.3%

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

   1. sub-neg 7.3%

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

5. Simplified 7.3%

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

   1. add-sqr-sqrt 10.6%

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

   2. pow2 10.6%

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

7. Applied egg-rr 10.6%

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

8. Final simplification 10.6%

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

Alternative 9: 6.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;{\left(\sqrt[3]{t_0}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\pi - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= (- 1.0 x) 1.0) (pow (cbrt t_0) 3.0) (- PI t_0))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = pow(cbrt(t_0), 3.0);
	} else {
		tmp = ((double) M_PI) - t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = Math.pow(Math.cbrt(t_0), 3.0);
	} else {
		tmp = Math.PI - t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = cbrt(t_0) ^ 3.0;
	else
		tmp = Float64(pi - t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision], N[(Pi - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 x) < 1

   1. Initial program 7.3%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. add-cube-cbrt 7.3%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\cos^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\cos^{-1} \left(1 - x\right)}} \]

      2. pow3 7.3%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{3}} \]

   3. Applied egg-rr 7.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{3}} \]

   if 1 < (-.f64 1 x)

   1. Initial program 7.3%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 7.3%

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

      2. sub-neg 7.3%

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

      3. div-inv 7.3%

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

      4. metadata-eval 7.3%

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

   3. Applied egg-rr 7.3%

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

      1. sub-neg 7.3%

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

   5. Simplified 7.3%

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

   6. Taylor expanded in x around 0 7.3%

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

      1. fma-neg 7.3%

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

      2. rem-square-sqrt 0.0%

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

      3. fabs-sqr 0.0%

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

      4. rem-square-sqrt 6.9%

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

      5. fabs-neg 6.9%

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

      6. rem-square-sqrt 6.9%

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

      7. fabs-sqr 6.9%

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

      8. rem-square-sqrt 6.9%

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

   8. Simplified 6.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \pi, \sin^{-1} \left(1 - x\right)\right)} \]
   9. Step-by-step derivation

      1. fma-udef 6.9%

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

      2. *-commutative 6.9%

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

      3. asin-acos 6.9%

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

      4. div-inv 6.9%

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

      5. metadata-eval 6.9%

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

      6. associate-+r- 6.9%

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

   10. Applied egg-rr 6.9%

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

       1. distribute-lft-out 6.9%

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

       2. metadata-eval 6.9%

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

       3. *-rgt-identity 6.9%

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

   12. Simplified 6.9%

       \[\leadsto \color{blue}{\pi - \cos^{-1} \left(1 - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \end{array} \]

Alternative 10: 6.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\log \left(e^{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= (- 1.0 x) 1.0) (log (exp t_0)) (- PI t_0))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = log(exp(t_0));
	} else {
		tmp = ((double) M_PI) - t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = Math.log(Math.exp(t_0));
	} else {
		tmp = Math.PI - t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if (1.0 - x) <= 1.0:
		tmp = math.log(math.exp(t_0))
	else:
		tmp = math.pi - t_0
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = log(exp(t_0));
	else
		tmp = Float64(pi - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if ((1.0 - x) <= 1.0)
		tmp = log(exp(t_0));
	else
		tmp = pi - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision], N[(Pi - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 x) < 1

   1. Initial program 7.3%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. add-log-exp 7.3%

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

   3. Applied egg-rr 7.3%

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

   if 1 < (-.f64 1 x)

   1. Initial program 7.3%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 7.3%

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

      2. sub-neg 7.3%

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

      3. div-inv 7.3%

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

      4. metadata-eval 7.3%

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

   3. Applied egg-rr 7.3%

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

      1. sub-neg 7.3%

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

   5. Simplified 7.3%

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

   6. Taylor expanded in x around 0 7.3%

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

      1. fma-neg 7.3%

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

      2. rem-square-sqrt 0.0%

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

      3. fabs-sqr 0.0%

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

      4. rem-square-sqrt 6.9%

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

      5. fabs-neg 6.9%

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

      6. rem-square-sqrt 6.9%

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

      7. fabs-sqr 6.9%

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

      8. rem-square-sqrt 6.9%

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

   8. Simplified 6.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \pi, \sin^{-1} \left(1 - x\right)\right)} \]
   9. Step-by-step derivation

      1. fma-udef 6.9%

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

      2. *-commutative 6.9%

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

      3. asin-acos 6.9%

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

      4. div-inv 6.9%

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

      5. metadata-eval 6.9%

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

      6. associate-+r- 6.9%

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

   10. Applied egg-rr 6.9%

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

       1. distribute-lft-out 6.9%

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

       2. metadata-eval 6.9%

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

       3. *-rgt-identity 6.9%

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

   12. Simplified 6.9%

       \[\leadsto \color{blue}{\pi - \cos^{-1} \left(1 - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \end{array} \]

Alternative 11: 6.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;-1 + \left(1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\pi - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= (- 1.0 x) 1.0) (+ -1.0 (+ 1.0 t_0)) (- PI t_0))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = -1.0 + (1.0 + t_0);
	} else {
		tmp = ((double) M_PI) - t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = -1.0 + (1.0 + t_0);
	} else {
		tmp = Math.PI - t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if (1.0 - x) <= 1.0:
		tmp = -1.0 + (1.0 + t_0)
	else:
		tmp = math.pi - t_0
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = Float64(-1.0 + Float64(1.0 + t_0));
	else
		tmp = Float64(pi - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if ((1.0 - x) <= 1.0)
		tmp = -1.0 + (1.0 + t_0);
	else
		tmp = pi - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 - x), $MachinePrecision], 1.0], N[(-1.0 + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(Pi - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 x) < 1

   1. Initial program 7.3%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. expm1-log1p-u 7.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      2. expm1-udef 7.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      3. log1p-udef 7.3%

         \[\leadsto e^{\color{blue}{\log \left(1 + \cos^{-1} \left(1 - x\right)\right)}} - 1 \]

      4. add-exp-log 7.3%

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

   3. Applied egg-rr 7.3%

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

   if 1 < (-.f64 1 x)

   1. Initial program 7.3%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin 7.3%

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

      2. sub-neg 7.3%

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

      3. div-inv 7.3%

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

      4. metadata-eval 7.3%

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

   3. Applied egg-rr 7.3%

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

      1. sub-neg 7.3%

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

   5. Simplified 7.3%

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

   6. Taylor expanded in x around 0 7.3%

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

      1. fma-neg 7.3%

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

      2. rem-square-sqrt 0.0%

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

      3. fabs-sqr 0.0%

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

      4. rem-square-sqrt 6.9%

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

      5. fabs-neg 6.9%

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

      6. rem-square-sqrt 6.9%

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

      7. fabs-sqr 6.9%

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

      8. rem-square-sqrt 6.9%

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

   8. Simplified 6.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \pi, \sin^{-1} \left(1 - x\right)\right)} \]
   9. Step-by-step derivation

      1. fma-udef 6.9%

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

      2. *-commutative 6.9%

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

      3. asin-acos 6.9%

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

      4. div-inv 6.9%

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

      5. metadata-eval 6.9%

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

      6. associate-+r- 6.9%

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

   10. Applied egg-rr 6.9%

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

       1. distribute-lft-out 6.9%

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

       2. metadata-eval 6.9%

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

       3. *-rgt-identity 6.9%

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

   12. Simplified 6.9%

       \[\leadsto \color{blue}{\pi - \cos^{-1} \left(1 - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;-1 + \left(1 + \cos^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi - \cos^{-1} \left(1 - x\right)\\ \end{array} \]

Alternative 12: 6.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \left(\cos^{-1} \left(1 - x\right) + -1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (+ (acos (- 1.0 x)) -1.0)))

C

double code(double x) {
	return 1.0 + (acos((1.0 - x)) + -1.0);
}

Fortran

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

Java

public static double code(double x) {
	return 1.0 + (Math.acos((1.0 - x)) + -1.0);
}

Python

def code(x):
	return 1.0 + (math.acos((1.0 - x)) + -1.0)

Julia

function code(x)
	return Float64(1.0 + Float64(acos(Float64(1.0 - x)) + -1.0))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + (acos((1.0 - x)) + -1.0);
end

Wolfram

code[x_] := N[(1.0 + N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. add-cube-cbrt 7.3%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\cos^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\cos^{-1} \left(1 - x\right)}} \]

   2. pow3 7.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{3}} \]

3. Applied egg-rr 7.3%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{3}} \]
4. Step-by-step derivation

   1. rem-cube-cbrt 7.3%

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

   2. expm1-log1p-u 7.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

   3. expm1-udef 7.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

   4. log1p-udef 7.3%

      \[\leadsto e^{\color{blue}{\log \left(1 + \cos^{-1} \left(1 - x\right)\right)}} - 1 \]

   5. *-rgt-identity 7.3%

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

   6. add-exp-log 7.3%

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

   7. *-rgt-identity 7.3%

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

   8. associate--l+ 7.3%

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

   9. +-commutative 7.3%

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

   10. sub-neg 7.3%

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

   11. metadata-eval 7.3%

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

5. Applied egg-rr 7.3%

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

6. Final simplification 7.3%

   \[\leadsto 1 + \left(\cos^{-1} \left(1 - x\right) + -1\right) \]

Alternative 13: 6.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -1 + \left(1 + \cos^{-1} \left(1 - x\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ -1.0 (+ 1.0 (acos (- 1.0 x)))))

C

double code(double x) {
	return -1.0 + (1.0 + acos((1.0 - x)));
}

Fortran

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

Java

public static double code(double x) {
	return -1.0 + (1.0 + Math.acos((1.0 - x)));
}

Python

def code(x):
	return -1.0 + (1.0 + math.acos((1.0 - x)))

Julia

function code(x)
	return Float64(-1.0 + Float64(1.0 + acos(Float64(1.0 - x))))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 + (1.0 + acos((1.0 - x)));
end

Wolfram

code[x_] := N[(-1.0 + N[(1.0 + N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. expm1-log1p-u 7.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

   2. expm1-udef 7.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

   3. log1p-udef 7.3%

      \[\leadsto e^{\color{blue}{\log \left(1 + \cos^{-1} \left(1 - x\right)\right)}} - 1 \]

   4. add-exp-log 7.3%

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

3. Applied egg-rr 7.3%

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

4. Final simplification 7.3%

   \[\leadsto -1 + \left(1 + \cos^{-1} \left(1 - x\right)\right) \]

Alternative 14: 6.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]

2. Final simplification 7.3%

   \[\leadsto \cos^{-1} \left(1 - x\right) \]

Developer target: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * asin(sqrt((x / 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023195 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :herbie-target
  (* 2.0 (asin (sqrt (/ x 2.0))))

  (acos (- 1.0 x)))