bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.4%

Time: 16.7s

Alternatives: 7

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: 6.9% 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: 10.4% 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 8.3%

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

   1. acos-asin 8.3%

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

   2. add-sqr-sqrt 6.5%

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

   3. fma-neg 6.5%

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

   4. div-inv 6.5%

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

   5. metadata-eval 6.5%

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

   6. div-inv 6.5%

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

   7. metadata-eval 6.5%

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

3. Applied egg-rr 6.5%

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

   1. sqrt-prod 11.7%

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

5. Applied egg-rr 11.7%

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

6. Final simplification 11.7%

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

Alternative 2: 10.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 + N[Log[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 8.3%

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

   1. expm1-log1p-u 8.3%

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

   2. expm1-udef 8.3%

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

   3. log1p-udef 8.3%

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

   4. rem-exp-log 8.3%

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

3. Applied egg-rr 8.3%

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

   1. associate--l+ 8.3%

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

   2. +-commutative 8.3%

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

   3. sub-neg 8.3%

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

   4. metadata-eval 8.3%

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

5. Applied egg-rr 8.3%

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

   1. log1p-expm1-u 8.3%

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

   2. log1p-udef 8.3%

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

7. Applied egg-rr 8.3%

   \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\cos^{-1} \left(1 - x\right) + -1\right)\right)} + 1 \]
8. Step-by-step derivation

   1. add-cbrt-cube 8.3%

      \[\leadsto \log \color{blue}{\left(\sqrt[3]{\left(\left(1 + \mathsf{expm1}\left(\cos^{-1} \left(1 - x\right) + -1\right)\right) \cdot \left(1 + \mathsf{expm1}\left(\cos^{-1} \left(1 - x\right) + -1\right)\right)\right) \cdot \left(1 + \mathsf{expm1}\left(\cos^{-1} \left(1 - x\right) + -1\right)\right)}\right)} + 1 \]

   2. pow1/3 11.6%

      \[\leadsto \log \color{blue}{\left({\left(\left(\left(1 + \mathsf{expm1}\left(\cos^{-1} \left(1 - x\right) + -1\right)\right) \cdot \left(1 + \mathsf{expm1}\left(\cos^{-1} \left(1 - x\right) + -1\right)\right)\right) \cdot \left(1 + \mathsf{expm1}\left(\cos^{-1} \left(1 - x\right) + -1\right)\right)\right)}^{0.3333333333333333}\right)} + 1 \]

   3. pow3 11.6%

      \[\leadsto \log \left({\color{blue}{\left({\left(1 + \mathsf{expm1}\left(\cos^{-1} \left(1 - x\right) + -1\right)\right)}^{3}\right)}}^{0.3333333333333333}\right) + 1 \]

   4. add-exp-log 11.6%

      \[\leadsto \log \left({\left({\color{blue}{\left(e^{\log \left(1 + \mathsf{expm1}\left(\cos^{-1} \left(1 - x\right) + -1\right)\right)}\right)}}^{3}\right)}^{0.3333333333333333}\right) + 1 \]

   5. log1p-def 11.6%

      \[\leadsto \log \left({\left({\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos^{-1} \left(1 - x\right) + -1\right)\right)}}\right)}^{3}\right)}^{0.3333333333333333}\right) + 1 \]

   6. log1p-expm1-u 11.6%

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

9. Applied egg-rr 11.6%

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

10. Final simplification 11.6%

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

Alternative 3: 10.3% 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 8.3%

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

   1. acos-asin 8.3%

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

   2. add-sqr-sqrt 6.5%

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

   3. fma-neg 6.5%

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

   4. div-inv 6.5%

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

   5. metadata-eval 6.5%

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

   6. div-inv 6.5%

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

   7. metadata-eval 6.5%

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

3. Applied egg-rr 6.5%

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

4. Taylor expanded in x around 0 11.6%

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

   1. add-cube-cbrt 11.6%

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

   2. pow3 11.6%

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

6. Applied egg-rr 11.6%

   \[\leadsto \pi \cdot {\left(\sqrt{0.5}\right)}^{2} - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]
7. Step-by-step derivation

   1. sqrt-pow2 11.6%

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

   2. metadata-eval 11.6%

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

   3. metadata-eval 11.6%

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

8. Applied egg-rr 11.6%

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

9. Final simplification 11.6%

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

Alternative 4: 10.3% 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 8.3%

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

   1. acos-asin 8.3%

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

   2. add-sqr-sqrt 6.5%

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

   3. fma-neg 6.5%

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

   4. div-inv 6.5%

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

   5. metadata-eval 6.5%

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

   6. div-inv 6.5%

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

   7. metadata-eval 6.5%

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

3. Applied egg-rr 6.5%

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

4. Taylor expanded in x around 0 11.6%

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

5. Final simplification 11.6%

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

Alternative 5: 6.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt{{\left(t_0 + -1\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= (- 1.0 x) 1.0) t_0 (+ 1.0 (sqrt (pow (+ t_0 -1.0) 2.0))))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 + sqrt(pow((t_0 + -1.0), 2.0));
	}
	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))
    if ((1.0d0 - x) <= 1.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0 + sqrt(((t_0 + (-1.0d0)) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 + Math.sqrt(Math.pow((t_0 + -1.0), 2.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if (1.0 - x) <= 1.0:
		tmp = t_0
	else:
		tmp = 1.0 + math.sqrt(math.pow((t_0 + -1.0), 2.0))
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 + sqrt((Float64(t_0 + -1.0) ^ 2.0)));
	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 = t_0;
	else
		tmp = 1.0 + sqrt(((t_0 + -1.0) ^ 2.0));
	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], t$95$0, N[(1.0 + N[Sqrt[N[Power[N[(t$95$0 + -1.0), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.3%

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

   if 1 < (-.f64 1 x)

   1. Initial program 8.3%

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

      1. expm1-log1p-u 8.3%

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

      2. expm1-udef 8.3%

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

      3. log1p-udef 8.3%

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

      4. rem-exp-log 8.3%

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

   3. Applied egg-rr 8.3%

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

      1. associate--l+ 8.3%

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

      2. +-commutative 8.3%

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

      3. sub-neg 8.3%

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

      4. metadata-eval 8.3%

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

   5. Applied egg-rr 8.3%

      \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1} \]
   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 7.0%

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

      3. pow2 7.0%

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

   7. Applied egg-rr 7.0%

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

4. Final simplification 8.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt{{\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2}}\\ \end{array} \]

Alternative 6: 6.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 1.0) (acos (- 1.0 x)) (+ (* PI 0.5) (asin (- 1.0 x)))))

C

double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = acos((1.0 - x));
	} else {
		tmp = (((double) M_PI) * 0.5) + asin((1.0 - x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - x) <= 1.0) {
		tmp = Math.acos((1.0 - x));
	} else {
		tmp = (Math.PI * 0.5) + Math.asin((1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 - x) <= 1.0:
		tmp = math.acos((1.0 - x))
	else:
		tmp = (math.pi * 0.5) + math.asin((1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - x) <= 1.0)
		tmp = acos(Float64(1.0 - x));
	else
		tmp = Float64(Float64(pi * 0.5) + asin(Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 - x) <= 1.0)
		tmp = acos((1.0 - x));
	else
		tmp = (pi * 0.5) + asin((1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.3%

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

   if 1 < (-.f64 1 x)

   1. Initial program 8.3%

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

      1. expm1-log1p-u 8.3%

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

      2. expm1-udef 8.3%

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

      3. log1p-udef 8.3%

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

      4. rem-exp-log 8.3%

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

   3. Applied egg-rr 8.3%

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

      1. add-exp-log 8.3%

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

      2. log1p-udef 8.3%

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

      3. expm1-udef 8.3%

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

      4. acos-asin 8.3%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)}\right)\right) \]

      5. expm1-log1p-u 8.3%

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

      6. div-inv 8.3%

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

      7. metadata-eval 8.3%

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

      8. add-sqr-sqrt 11.6%

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

      9. cancel-sign-sub-inv 11.6%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 7.0%

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

      12. sqr-neg 7.0%

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

      13. add-sqr-sqrt 7.0%

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

   5. Applied egg-rr 7.0%

      \[\leadsto \color{blue}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 8.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)\\ \end{array} \]

Alternative 7: 6.9% 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 8.3%

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

2. Final simplification 8.3%

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

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 2023322 
(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)))