bug323 (missed optimization)

Percentage Accurate: 7.0% → 10.6%

Time: 10.2s

Alternatives: 9

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: 7.0% 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: 10.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

   1. add-log-exp 7.0%

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

3. Applied egg-rr 7.0%

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

   1. add-log-exp 7.0%

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

   2. acos-asin 7.0%

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

   3. div-inv 7.0%

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

   4. metadata-eval 7.0%

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

   5. add-sqr-sqrt 5.2%

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

   6. fma-neg 5.2%

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

5. Applied egg-rr 5.2%

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

   1. sqrt-prod 10.8%

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

7. Applied egg-rr 10.8%

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

8. Final simplification 10.8%

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

Alternative 2: 10.5% 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 7.0%

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

   1. acos-asin 7.0%

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

   2. sub-neg 7.0%

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

   3. div-inv 7.0%

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

   4. metadata-eval 7.0%

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

3. Applied egg-rr 7.0%

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

   1. sub-neg 7.0%

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

5. Simplified 7.0%

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

   1. add-cube-cbrt 10.7%

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

   2. pow3 10.7%

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

7. Applied egg-rr 10.7%

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

8. Final simplification 10.7%

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

Alternative 3: 10.5% 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 7.0%

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

   1. acos-asin 7.0%

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

   2. sub-neg 7.0%

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

   3. div-inv 7.0%

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

   4. metadata-eval 7.0%

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

3. Applied egg-rr 7.0%

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

   1. sub-neg 7.0%

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

5. Simplified 7.0%

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

   1. add-sqr-sqrt 10.7%

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

   2. pow2 10.7%

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

7. Applied egg-rr 10.7%

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

8. Final simplification 10.7%

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

Alternative 4: 10.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

   1. add-log-exp 7.0%

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

3. Applied egg-rr 7.0%

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

   1. add-log-exp 7.0%

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

   2. acos-asin 7.0%

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

   3. div-inv 7.0%

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

   4. metadata-eval 7.0%

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

   5. add-sqr-sqrt 5.2%

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

   6. fma-neg 5.2%

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

5. Applied egg-rr 5.2%

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

6. Taylor expanded in x around 0 10.8%

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

7. Final simplification 10.8%

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

Alternative 5: 9.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right) + -1\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;1 + \left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;1 + {\left(\sqrt[3]{t_0}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (acos (- 1.0 x)) -1.0)))
   (if (<= x 5.5e-17) (+ 1.0 (fabs t_0)) (+ 1.0 (pow (cbrt t_0) 3.0)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x)) + -1.0;
	double tmp;
	if (x <= 5.5e-17) {
		tmp = 1.0 + fabs(t_0);
	} else {
		tmp = 1.0 + pow(cbrt(t_0), 3.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x)) + -1.0;
	double tmp;
	if (x <= 5.5e-17) {
		tmp = 1.0 + Math.abs(t_0);
	} else {
		tmp = 1.0 + Math.pow(Math.cbrt(t_0), 3.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(acos(Float64(1.0 - x)) + -1.0)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(1.0 + abs(t_0));
	else
		tmp = Float64(1.0 + (cbrt(t_0) ^ 3.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(1.0 + N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.8%

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

      1. add-log-exp 3.8%

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

   3. Applied egg-rr 3.8%

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

      1. add-log-exp 3.8%

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

      2. expm1-log1p-u 3.8%

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

      3. expm1-udef 3.8%

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

      4. log1p-udef 3.8%

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

      5. add-exp-log 3.8%

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

      6. associate--l+ 3.8%

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

      7. +-commutative 3.8%

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

      8. sub-neg 3.8%

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

      9. metadata-eval 3.8%

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

   5. Applied egg-rr 3.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.7%

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

      3. pow2 6.7%

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

   7. Applied egg-rr 6.7%

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

      1. unpow2 6.7%

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

      2. rem-sqrt-square 6.7%

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

   9. Simplified 6.7%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 72.5%

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

      1. add-log-exp 72.6%

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

   3. Applied egg-rr 72.6%

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

      1. add-log-exp 72.5%

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

      2. expm1-log1p-u 72.4%

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

      3. expm1-udef 72.4%

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

      4. log1p-udef 72.4%

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

      5. add-exp-log 72.4%

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

      6. associate--l+ 72.5%

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

      7. +-commutative 72.5%

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

      8. sub-neg 72.5%

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

      9. metadata-eval 72.5%

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

   5. Applied egg-rr 72.5%

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

      1. add-cube-cbrt 73.1%

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

      2. pow3 73.2%

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

   7. Applied egg-rr 73.2%

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

4. Final simplification 9.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;1 + \left|\cos^{-1} \left(1 - x\right) + -1\right|\\ \mathbf{else}:\\ \;\;\;\;1 + {\left(\sqrt[3]{\cos^{-1} \left(1 - x\right) + -1}\right)}^{3}\\ \end{array} \]

Alternative 6: 9.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right) + -1\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;1 + \left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;1 + 3 \cdot \left(t_0 \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (acos (- 1.0 x)) -1.0)))
   (if (<= x 5.5e-17)
     (+ 1.0 (fabs t_0))
     (+ 1.0 (* 3.0 (* t_0 0.3333333333333333))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x)) + -1.0;
	double tmp;
	if (x <= 5.5e-17) {
		tmp = 1.0 + fabs(t_0);
	} else {
		tmp = 1.0 + (3.0 * (t_0 * 0.3333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = acos((1.0d0 - x)) + (-1.0d0)
    if (x <= 5.5d-17) then
        tmp = 1.0d0 + abs(t_0)
    else
        tmp = 1.0d0 + (3.0d0 * (t_0 * 0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x)) + -1.0;
	double tmp;
	if (x <= 5.5e-17) {
		tmp = 1.0 + Math.abs(t_0);
	} else {
		tmp = 1.0 + (3.0 * (t_0 * 0.3333333333333333));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x)) + -1.0
	tmp = 0
	if x <= 5.5e-17:
		tmp = 1.0 + math.fabs(t_0)
	else:
		tmp = 1.0 + (3.0 * (t_0 * 0.3333333333333333))
	return tmp

Julia

function code(x)
	t_0 = Float64(acos(Float64(1.0 - x)) + -1.0)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(1.0 + abs(t_0));
	else
		tmp = Float64(1.0 + Float64(3.0 * Float64(t_0 * 0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x)) + -1.0;
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = 1.0 + abs(t_0);
	else
		tmp = 1.0 + (3.0 * (t_0 * 0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(1.0 + N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(3.0 * N[(t$95$0 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.8%

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

      1. add-log-exp 3.8%

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

   3. Applied egg-rr 3.8%

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

      1. add-log-exp 3.8%

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

      2. expm1-log1p-u 3.8%

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

      3. expm1-udef 3.8%

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

      4. log1p-udef 3.8%

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

      5. add-exp-log 3.8%

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

      6. associate--l+ 3.8%

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

      7. +-commutative 3.8%

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

      8. sub-neg 3.8%

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

      9. metadata-eval 3.8%

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

   5. Applied egg-rr 3.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.7%

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

      3. pow2 6.7%

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

   7. Applied egg-rr 6.7%

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

      1. unpow2 6.7%

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

      2. rem-sqrt-square 6.7%

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

   9. Simplified 6.7%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 72.5%

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

      1. add-log-exp 72.6%

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

   3. Applied egg-rr 72.6%

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

      1. add-log-exp 72.5%

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

      2. expm1-log1p-u 72.4%

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

      3. expm1-udef 72.4%

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

      4. log1p-udef 72.4%

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

      5. add-exp-log 72.4%

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

      6. associate--l+ 72.5%

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

      7. +-commutative 72.5%

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

      8. sub-neg 72.5%

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

      9. metadata-eval 72.5%

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

   5. Applied egg-rr 72.5%

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

      1. add-cube-cbrt 73.1%

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

      2. pow3 73.2%

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

   7. Applied egg-rr 73.2%

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

      1. rem-cube-cbrt 72.5%

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

      2. add-log-exp 72.5%

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

      3. add-cube-cbrt 71.8%

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

      4. unpow3 71.7%

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

      5. pow-to-exp 72.1%

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

      6. add-log-exp 71.8%

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

      7. rem-cbrt-cube 71.5%

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

      8. pow1/3 71.8%

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

      9. log-pow 71.9%

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

      10. unpow3 71.9%

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

      11. add-cube-cbrt 72.7%

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

      12. add-log-exp 72.7%

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

   9. Applied egg-rr 72.7%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(\cos^{-1} \left(1 - x\right) + -1\right)\right) \cdot 3} + 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;1 + \left|\cos^{-1} \left(1 - x\right) + -1\right|\\ \mathbf{else}:\\ \;\;\;\;1 + 3 \cdot \left(\left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot 0.3333333333333333\right)\\ \end{array} \]

Alternative 7: 9.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi + t_0\\ \mathbf{else}:\\ \;\;\;\;1 + 3 \cdot \left(\left(t_0 + -1\right) \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= x 5.5e-17)
     (+ PI t_0)
     (+ 1.0 (* 3.0 (* (+ t_0 -1.0) 0.3333333333333333))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = ((double) M_PI) + t_0;
	} else {
		tmp = 1.0 + (3.0 * ((t_0 + -1.0) * 0.3333333333333333));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = Math.PI + t_0;
	} else {
		tmp = 1.0 + (3.0 * ((t_0 + -1.0) * 0.3333333333333333));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.pi + t_0
	else:
		tmp = 1.0 + (3.0 * ((t_0 + -1.0) * 0.3333333333333333))
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(pi + t_0);
	else
		tmp = Float64(1.0 + Float64(3.0 * Float64(Float64(t_0 + -1.0) * 0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = pi + t_0;
	else
		tmp = 1.0 + (3.0 * ((t_0 + -1.0) * 0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(Pi + t$95$0), $MachinePrecision], N[(1.0 + N[(3.0 * N[(N[(t$95$0 + -1.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.8%

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

      1. acos-asin 3.8%

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

      2. sub-neg 3.8%

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

      3. div-inv 3.8%

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

      4. metadata-eval 3.8%

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

   3. Applied egg-rr 3.8%

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

      1. sub-neg 3.8%

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

   5. Simplified 3.8%

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

      1. asin-acos 3.8%

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

      2. div-inv 3.8%

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

      3. metadata-eval 3.8%

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

      4. add-sqr-sqrt 7.7%

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

      5. fma-neg 7.7%

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

   7. Applied egg-rr 7.7%

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

      1. fma-udef 7.7%

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

      2. add-sqr-sqrt 3.8%

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

      3. sub-neg 3.8%

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

      4. metadata-eval 3.8%

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

      5. div-inv 3.8%

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

      6. asin-acos 3.8%

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

      7. sub-neg 3.8%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 6.7%

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

      10. sqr-neg 6.7%

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

      11. sqrt-unprod 6.7%

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

      12. add-sqr-sqrt 6.7%

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

      13. asin-acos 6.7%

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

      14. div-inv 6.7%

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

      15. metadata-eval 6.7%

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

      16. sub-neg 6.7%

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

   9. Applied egg-rr 6.7%

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

       1. +-commutative 6.7%

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

       2. distribute-lft-out 6.7%

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

       3. metadata-eval 6.7%

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

       4. *-rgt-identity 6.7%

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

   11. Simplified 6.7%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 72.5%

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

      1. add-log-exp 72.6%

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

   3. Applied egg-rr 72.6%

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

      1. add-log-exp 72.5%

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

      2. expm1-log1p-u 72.4%

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

      3. expm1-udef 72.4%

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

      4. log1p-udef 72.4%

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

      5. add-exp-log 72.4%

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

      6. associate--l+ 72.5%

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

      7. +-commutative 72.5%

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

      8. sub-neg 72.5%

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

      9. metadata-eval 72.5%

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

   5. Applied egg-rr 72.5%

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

      1. add-cube-cbrt 73.1%

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

      2. pow3 73.2%

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

   7. Applied egg-rr 73.2%

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

      1. rem-cube-cbrt 72.5%

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

      2. add-log-exp 72.5%

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

      3. add-cube-cbrt 71.8%

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

      4. unpow3 71.7%

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

      5. pow-to-exp 72.1%

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

      6. add-log-exp 71.8%

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

      7. rem-cbrt-cube 71.5%

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

      8. pow1/3 71.8%

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

      9. log-pow 71.9%

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

      10. unpow3 71.9%

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

      11. add-cube-cbrt 72.7%

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

      12. add-log-exp 72.7%

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

   9. Applied egg-rr 72.7%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(\cos^{-1} \left(1 - x\right) + -1\right)\right) \cdot 3} + 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\pi + \cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 3 \cdot \left(\left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot 0.3333333333333333\right)\\ \end{array} \]

Alternative 8: 7.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (+ 1.0 (* 3.0 (* (+ (acos (- 1.0 x)) -1.0) 0.3333333333333333))))

C

double code(double x) {
	return 1.0 + (3.0 * ((acos((1.0 - x)) + -1.0) * 0.3333333333333333));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + (3.0d0 * ((acos((1.0d0 - x)) + (-1.0d0)) * 0.3333333333333333d0))
end function

Java

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

Python

def code(x):
	return 1.0 + (3.0 * ((math.acos((1.0 - x)) + -1.0) * 0.3333333333333333))

Julia

function code(x)
	return Float64(1.0 + Float64(3.0 * Float64(Float64(acos(Float64(1.0 - x)) + -1.0) * 0.3333333333333333)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

   1. add-log-exp 7.0%

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

3. Applied egg-rr 7.0%

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

   1. add-log-exp 7.0%

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

   2. expm1-log1p-u 7.0%

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

   3. expm1-udef 7.0%

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

   4. log1p-udef 7.0%

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

   5. add-exp-log 7.0%

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

   6. associate--l+ 7.0%

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

   7. +-commutative 7.0%

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

   8. sub-neg 7.0%

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

   9. metadata-eval 7.0%

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

5. Applied egg-rr 7.0%

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

   1. add-cube-cbrt 7.1%

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

   2. pow3 7.1%

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

7. Applied egg-rr 7.1%

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

   1. rem-cube-cbrt 7.0%

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

   2. add-log-exp 7.0%

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

   3. add-cube-cbrt 5.2%

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

   4. unpow3 5.2%

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

   5. pow-to-exp 5.2%

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

   6. add-log-exp 5.2%

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

   7. rem-cbrt-cube 5.2%

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

   8. pow1/3 5.2%

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

   9. log-pow 5.2%

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

   10. unpow3 5.2%

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

   11. add-cube-cbrt 7.1%

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

   12. add-log-exp 7.1%

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

9. Applied egg-rr 7.1%

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

10. Final simplification 7.1%

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

Alternative 9: 7.0% 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 7.0%

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

2. Final simplification 7.0%

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

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 2023293 
(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)))