VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.0%

Time: 17.1s

Alternatives: 7

Speedup: 4.9×

• could not determine a ground truth

  1. f = 1474397023.13528

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t\_0}\\ t_2 := e^{-t\_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t\_1 + t\_2}{t\_1 - t\_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Initial Program: 6.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t\_0}\\ t_2 := e^{-t\_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t\_1 + t\_2}{t\_1 - t\_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Alternative 1: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sqrt{\pi}}\\ -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\left({\left(t\_0 \cdot t\_0\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot 0.5\right)\right)} + \left(-1 + \frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right)\right)}{\pi} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (cbrt (sqrt PI))))
   (*
    -4.0
    (/
     (log1p
      (+
       (/ 1.0 (expm1 (* f (* (* (pow (* t_0 t_0) 2.0) (cbrt PI)) 0.5))))
       (+ -1.0 (/ -1.0 (expm1 (* f (* PI -0.5)))))))
     PI))))

C

double code(double f) {
	double t_0 = cbrt(sqrt(((double) M_PI)));
	return -4.0 * (log1p(((1.0 / expm1((f * ((pow((t_0 * t_0), 2.0) * cbrt(((double) M_PI))) * 0.5)))) + (-1.0 + (-1.0 / expm1((f * (((double) M_PI) * -0.5))))))) / ((double) M_PI));
}

Java

public static double code(double f) {
	double t_0 = Math.cbrt(Math.sqrt(Math.PI));
	return -4.0 * (Math.log1p(((1.0 / Math.expm1((f * ((Math.pow((t_0 * t_0), 2.0) * Math.cbrt(Math.PI)) * 0.5)))) + (-1.0 + (-1.0 / Math.expm1((f * (Math.PI * -0.5))))))) / Math.PI);
}

Julia

function code(f)
	t_0 = cbrt(sqrt(pi))
	return Float64(-4.0 * Float64(log1p(Float64(Float64(1.0 / expm1(Float64(f * Float64(Float64((Float64(t_0 * t_0) ^ 2.0) * cbrt(pi)) * 0.5)))) + Float64(-1.0 + Float64(-1.0 / expm1(Float64(f * Float64(pi * -0.5))))))) / pi))
end

Wolfram

code[f_] := Block[{t$95$0 = N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision]}, N[(-4.0 * N[(N[Log[1 + N[(N[(1.0 / N[(Exp[N[(f * N[(N[(N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[Pi, 1/3], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[(-1.0 / N[(Exp[N[(f * N[(Pi * -0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

3. Simplified 98.8%

   \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around inf 6.8%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{e^{0.5 \cdot \left(f \cdot \pi\right)} - 1} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi}} \]
6. Step-by-step derivation

   1. expm1-define 7.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\color{blue}{\mathsf{expm1}\left(0.5 \cdot \left(f \cdot \pi\right)\right)}} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   2. *-commutative 7.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\left(f \cdot \pi\right) \cdot 0.5}\right)} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   3. associate-*l* 7.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\color{blue}{f \cdot \left(\pi \cdot 0.5\right)}\right)} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   4. expm1-define 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\color{blue}{\mathsf{expm1}\left(-0.5 \cdot \left(f \cdot \pi\right)\right)}}\right)}{\pi} \]

   5. associate-*r* 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\left(-0.5 \cdot f\right) \cdot \pi}\right)}\right)}{\pi} \]

   6. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\pi \cdot \left(-0.5 \cdot f\right)}\right)}\right)}{\pi} \]

   7. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \color{blue}{\left(f \cdot -0.5\right)}\right)}\right)}{\pi} \]

7. Simplified 98.9%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}} \]
8. Step-by-step derivation

   1. log1p-expm1-u 98.9%

      \[\leadsto -4 \cdot \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\right)\right)}}{\pi} \]

   2. expm1-undefine 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)} - 1}\right)}{\pi} \]

   3. add-exp-log 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)} - 1\right)}{\pi} \]

   4. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \color{blue}{\left(-0.5 \cdot f\right)}\right)}\right) - 1\right)}{\pi} \]

   5. associate-*r* 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\left(\pi \cdot -0.5\right) \cdot f}\right)}\right) - 1\right)}{\pi} \]

   6. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{f \cdot \left(\pi \cdot -0.5\right)}\right)}\right) - 1\right)}{\pi} \]

9. Applied egg-rr 98.9%

   \[\leadsto -4 \cdot \frac{\color{blue}{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right) - 1\right)}}{\pi} \]
10. Step-by-step derivation

    1. sub-neg 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right) + \left(-1\right)}\right)}{\pi} \]

    2. sub-neg 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \left(-\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right)\right)} + \left(-1\right)\right)}{\pi} \]

    3. distribute-neg-frac 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \color{blue}{\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}}\right) + \left(-1\right)\right)}{\pi} \]

    4. metadata-eval 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{\color{blue}{-1}}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right) + \left(-1\right)\right)}{\pi} \]

    5. metadata-eval 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right) + \color{blue}{-1}\right)}{\pi} \]

    6. associate-+l+ 99.0%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} + -1\right)}\right)}{\pi} \]

11. Simplified 99.0%

    \[\leadsto -4 \cdot \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} + -1\right)\right)}}{\pi} \]
12. Step-by-step derivation

    1. add-cube-cbrt 99.0%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} + -1\right)\right)}{\pi} \]

    2. pow2 99.0%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}\right) \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} + -1\right)\right)}{\pi} \]

13. Applied egg-rr 99.0%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}\right)} \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} + -1\right)\right)}{\pi} \]
14. Step-by-step derivation

    1. pow1/3 99.0%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\left({\color{blue}{\left({\pi}^{0.3333333333333333}\right)}}^{2} \cdot \sqrt[3]{\pi}\right) \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} + -1\right)\right)}{\pi} \]

    2. add-sqr-sqrt 99.0%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\left({\left({\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)}}^{0.3333333333333333}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} + -1\right)\right)}{\pi} \]

    3. unpow-prod-down 99.0%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\left({\color{blue}{\left({\left(\sqrt{\pi}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\pi}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \sqrt[3]{\pi}\right) \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} + -1\right)\right)}{\pi} \]

15. Applied egg-rr 99.0%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\left({\color{blue}{\left({\left(\sqrt{\pi}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\pi}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \sqrt[3]{\pi}\right) \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} + -1\right)\right)}{\pi} \]
16. Step-by-step derivation

    1. unpow1/3 99.0%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\left({\left(\color{blue}{\sqrt[3]{\sqrt{\pi}}} \cdot {\left(\sqrt{\pi}\right)}^{0.3333333333333333}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} + -1\right)\right)}{\pi} \]

    2. unpow1/3 99.0%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\left({\left(\sqrt[3]{\sqrt{\pi}} \cdot \color{blue}{\sqrt[3]{\sqrt{\pi}}}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} + -1\right)\right)}{\pi} \]

17. Simplified 99.0%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\left({\color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)}}^{2} \cdot \sqrt[3]{\pi}\right) \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} + -1\right)\right)}{\pi} \]

18. Final simplification 99.0%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\left({\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)}^{2} \cdot \sqrt[3]{\pi}\right) \cdot 0.5\right)\right)} + \left(-1 + \frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right)\right)}{\pi} \]
19. Add Preprocessing

Alternative 2: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\mathsf{log1p}\left(\left(-1 + \frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right) + \frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log1p
    (+
     (+ -1.0 (/ -1.0 (expm1 (* f (* PI -0.5)))))
     (/ 1.0 (expm1 (* f (* PI 0.5))))))
   PI)))

C

double code(double f) {
	return -4.0 * (log1p(((-1.0 + (-1.0 / expm1((f * (((double) M_PI) * -0.5))))) + (1.0 / expm1((f * (((double) M_PI) * 0.5)))))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log1p(((-1.0 + (-1.0 / Math.expm1((f * (Math.PI * -0.5))))) + (1.0 / Math.expm1((f * (Math.PI * 0.5)))))) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log1p(((-1.0 + (-1.0 / math.expm1((f * (math.pi * -0.5))))) + (1.0 / math.expm1((f * (math.pi * 0.5)))))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log1p(Float64(Float64(-1.0 + Float64(-1.0 / expm1(Float64(f * Float64(pi * -0.5))))) + Float64(1.0 / expm1(Float64(f * Float64(pi * 0.5)))))) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[1 + N[(N[(-1.0 + N[(-1.0 / N[(Exp[N[(f * N[(Pi * -0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(Exp[N[(f * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

3. Simplified 98.8%

   \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around inf 6.8%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{e^{0.5 \cdot \left(f \cdot \pi\right)} - 1} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi}} \]
6. Step-by-step derivation

   1. expm1-define 7.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\color{blue}{\mathsf{expm1}\left(0.5 \cdot \left(f \cdot \pi\right)\right)}} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   2. *-commutative 7.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\left(f \cdot \pi\right) \cdot 0.5}\right)} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   3. associate-*l* 7.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\color{blue}{f \cdot \left(\pi \cdot 0.5\right)}\right)} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   4. expm1-define 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\color{blue}{\mathsf{expm1}\left(-0.5 \cdot \left(f \cdot \pi\right)\right)}}\right)}{\pi} \]

   5. associate-*r* 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\left(-0.5 \cdot f\right) \cdot \pi}\right)}\right)}{\pi} \]

   6. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\pi \cdot \left(-0.5 \cdot f\right)}\right)}\right)}{\pi} \]

   7. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \color{blue}{\left(f \cdot -0.5\right)}\right)}\right)}{\pi} \]

7. Simplified 98.9%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}} \]
8. Step-by-step derivation

   1. log1p-expm1-u 98.9%

      \[\leadsto -4 \cdot \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\right)\right)}}{\pi} \]

   2. expm1-undefine 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)} - 1}\right)}{\pi} \]

   3. add-exp-log 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)} - 1\right)}{\pi} \]

   4. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \color{blue}{\left(-0.5 \cdot f\right)}\right)}\right) - 1\right)}{\pi} \]

   5. associate-*r* 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\left(\pi \cdot -0.5\right) \cdot f}\right)}\right) - 1\right)}{\pi} \]

   6. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{f \cdot \left(\pi \cdot -0.5\right)}\right)}\right) - 1\right)}{\pi} \]

9. Applied egg-rr 98.9%

   \[\leadsto -4 \cdot \frac{\color{blue}{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right) - 1\right)}}{\pi} \]
10. Step-by-step derivation

    1. sub-neg 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right) + \left(-1\right)}\right)}{\pi} \]

    2. sub-neg 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \left(-\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right)\right)} + \left(-1\right)\right)}{\pi} \]

    3. distribute-neg-frac 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \color{blue}{\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}}\right) + \left(-1\right)\right)}{\pi} \]

    4. metadata-eval 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{\color{blue}{-1}}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right) + \left(-1\right)\right)}{\pi} \]

    5. metadata-eval 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right) + \color{blue}{-1}\right)}{\pi} \]

    6. associate-+l+ 99.0%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} + -1\right)}\right)}{\pi} \]

11. Simplified 99.0%

    \[\leadsto -4 \cdot \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} + -1\right)\right)}}{\pi} \]

12. Final simplification 99.0%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(-1 + \frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right) + \frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)}\right)}{\pi} \]
13. Add Preprocessing

Alternative 3: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log
    (+ (/ 1.0 (expm1 (* f (* PI 0.5)))) (/ -1.0 (expm1 (* PI (* f -0.5))))))
   PI)))

C

double code(double f) {
	return -4.0 * (log(((1.0 / expm1((f * (((double) M_PI) * 0.5)))) + (-1.0 / expm1((((double) M_PI) * (f * -0.5)))))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log(((1.0 / Math.expm1((f * (Math.PI * 0.5)))) + (-1.0 / Math.expm1((Math.PI * (f * -0.5)))))) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log(((1.0 / math.expm1((f * (math.pi * 0.5)))) + (-1.0 / math.expm1((math.pi * (f * -0.5)))))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log(Float64(Float64(1.0 / expm1(Float64(f * Float64(pi * 0.5)))) + Float64(-1.0 / expm1(Float64(pi * Float64(f * -0.5)))))) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(N[(1.0 / N[(Exp[N[(f * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(Exp[N[(Pi * N[(f * -0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

3. Simplified 98.8%

   \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around inf 6.8%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{e^{0.5 \cdot \left(f \cdot \pi\right)} - 1} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi}} \]
6. Step-by-step derivation

   1. expm1-define 7.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\color{blue}{\mathsf{expm1}\left(0.5 \cdot \left(f \cdot \pi\right)\right)}} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   2. *-commutative 7.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\left(f \cdot \pi\right) \cdot 0.5}\right)} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   3. associate-*l* 7.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\color{blue}{f \cdot \left(\pi \cdot 0.5\right)}\right)} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   4. expm1-define 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\color{blue}{\mathsf{expm1}\left(-0.5 \cdot \left(f \cdot \pi\right)\right)}}\right)}{\pi} \]

   5. associate-*r* 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\left(-0.5 \cdot f\right) \cdot \pi}\right)}\right)}{\pi} \]

   6. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\pi \cdot \left(-0.5 \cdot f\right)}\right)}\right)}{\pi} \]

   7. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \color{blue}{\left(f \cdot -0.5\right)}\right)}\right)}{\pi} \]

7. Simplified 98.9%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}} \]

8. Final simplification 98.9%

   \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi} \]
9. Add Preprocessing

Alternative 4: 96.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{f \cdot \left(f \cdot \left(\pi \cdot 0.041666666666666664\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log1p
    (+
     (/ 1.0 (expm1 (* f (* PI 0.5))))
     (/
      (+ (* f (- (* f (* PI 0.041666666666666664)) 0.5)) (* 2.0 (/ 1.0 PI)))
      f)))
   PI)))

C

double code(double f) {
	return -4.0 * (log1p(((1.0 / expm1((f * (((double) M_PI) * 0.5)))) + (((f * ((f * (((double) M_PI) * 0.041666666666666664)) - 0.5)) + (2.0 * (1.0 / ((double) M_PI)))) / f))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log1p(((1.0 / Math.expm1((f * (Math.PI * 0.5)))) + (((f * ((f * (Math.PI * 0.041666666666666664)) - 0.5)) + (2.0 * (1.0 / Math.PI))) / f))) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log1p(((1.0 / math.expm1((f * (math.pi * 0.5)))) + (((f * ((f * (math.pi * 0.041666666666666664)) - 0.5)) + (2.0 * (1.0 / math.pi))) / f))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log1p(Float64(Float64(1.0 / expm1(Float64(f * Float64(pi * 0.5)))) + Float64(Float64(Float64(f * Float64(Float64(f * Float64(pi * 0.041666666666666664)) - 0.5)) + Float64(2.0 * Float64(1.0 / pi))) / f))) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[1 + N[(N[(1.0 / N[(Exp[N[(f * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(f * N[(N[(f * N[(Pi * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

3. Simplified 98.8%

   \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around inf 6.8%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{e^{0.5 \cdot \left(f \cdot \pi\right)} - 1} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi}} \]
6. Step-by-step derivation

   1. expm1-define 7.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\color{blue}{\mathsf{expm1}\left(0.5 \cdot \left(f \cdot \pi\right)\right)}} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   2. *-commutative 7.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\left(f \cdot \pi\right) \cdot 0.5}\right)} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   3. associate-*l* 7.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\color{blue}{f \cdot \left(\pi \cdot 0.5\right)}\right)} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   4. expm1-define 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\color{blue}{\mathsf{expm1}\left(-0.5 \cdot \left(f \cdot \pi\right)\right)}}\right)}{\pi} \]

   5. associate-*r* 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\left(-0.5 \cdot f\right) \cdot \pi}\right)}\right)}{\pi} \]

   6. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\pi \cdot \left(-0.5 \cdot f\right)}\right)}\right)}{\pi} \]

   7. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \color{blue}{\left(f \cdot -0.5\right)}\right)}\right)}{\pi} \]

7. Simplified 98.9%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}} \]
8. Step-by-step derivation

   1. log1p-expm1-u 98.9%

      \[\leadsto -4 \cdot \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\right)\right)}}{\pi} \]

   2. expm1-undefine 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)} - 1}\right)}{\pi} \]

   3. add-exp-log 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)} - 1\right)}{\pi} \]

   4. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \color{blue}{\left(-0.5 \cdot f\right)}\right)}\right) - 1\right)}{\pi} \]

   5. associate-*r* 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\left(\pi \cdot -0.5\right) \cdot f}\right)}\right) - 1\right)}{\pi} \]

   6. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{f \cdot \left(\pi \cdot -0.5\right)}\right)}\right) - 1\right)}{\pi} \]

9. Applied egg-rr 98.9%

   \[\leadsto -4 \cdot \frac{\color{blue}{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right) - 1\right)}}{\pi} \]
10. Step-by-step derivation

    1. sub-neg 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right) + \left(-1\right)}\right)}{\pi} \]

    2. sub-neg 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \left(-\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right)\right)} + \left(-1\right)\right)}{\pi} \]

    3. distribute-neg-frac 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \color{blue}{\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}}\right) + \left(-1\right)\right)}{\pi} \]

    4. metadata-eval 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{\color{blue}{-1}}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right) + \left(-1\right)\right)}{\pi} \]

    5. metadata-eval 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right) + \color{blue}{-1}\right)}{\pi} \]

    6. associate-+l+ 99.0%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} + -1\right)}\right)}{\pi} \]

11. Simplified 99.0%

    \[\leadsto -4 \cdot \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} + -1\right)\right)}}{\pi} \]

12. Taylor expanded in f around 0 96.7%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \color{blue}{\frac{f \cdot \left(f \cdot \left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}}\right)}{\pi} \]
13. Step-by-step derivation

    1. *-un-lft-identity 96.7%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{f \cdot \left(f \cdot \color{blue}{\left(1 \cdot \left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

    2. distribute-rgt-out 96.7%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{f \cdot \left(f \cdot \left(1 \cdot \color{blue}{\left(\pi \cdot \left(-0.08333333333333333 + 0.125\right)\right)}\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

    3. metadata-eval 96.7%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{f \cdot \left(f \cdot \left(1 \cdot \left(\pi \cdot \color{blue}{0.041666666666666664}\right)\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

14. Applied egg-rr 96.7%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{f \cdot \left(f \cdot \color{blue}{\left(1 \cdot \left(\pi \cdot 0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]
15. Step-by-step derivation

    1. *-lft-identity 96.7%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{f \cdot \left(f \cdot \color{blue}{\left(\pi \cdot 0.041666666666666664\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

16. Simplified 96.7%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{f \cdot \left(f \cdot \color{blue}{\left(\pi \cdot 0.041666666666666664\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]
17. Add Preprocessing

Alternative 5: 96.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(-1 + \pi \cdot \left(f \cdot 0.08333333333333333\right)\right) + \frac{1}{\pi} \cdot 4}{f}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log1p
    (/
     (+ (* f (+ -1.0 (* PI (* f 0.08333333333333333)))) (* (/ 1.0 PI) 4.0))
     f))
   PI)))

C

double code(double f) {
	return -4.0 * (log1p((((f * (-1.0 + (((double) M_PI) * (f * 0.08333333333333333)))) + ((1.0 / ((double) M_PI)) * 4.0)) / f)) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log1p((((f * (-1.0 + (Math.PI * (f * 0.08333333333333333)))) + ((1.0 / Math.PI) * 4.0)) / f)) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log1p((((f * (-1.0 + (math.pi * (f * 0.08333333333333333)))) + ((1.0 / math.pi) * 4.0)) / f)) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log1p(Float64(Float64(Float64(f * Float64(-1.0 + Float64(pi * Float64(f * 0.08333333333333333)))) + Float64(Float64(1.0 / pi) * 4.0)) / f)) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[1 + N[(N[(N[(f * N[(-1.0 + N[(Pi * N[(f * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / Pi), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

3. Simplified 98.8%

   \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around inf 6.8%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{e^{0.5 \cdot \left(f \cdot \pi\right)} - 1} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi}} \]
6. Step-by-step derivation

   1. expm1-define 7.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\color{blue}{\mathsf{expm1}\left(0.5 \cdot \left(f \cdot \pi\right)\right)}} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   2. *-commutative 7.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\left(f \cdot \pi\right) \cdot 0.5}\right)} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   3. associate-*l* 7.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\color{blue}{f \cdot \left(\pi \cdot 0.5\right)}\right)} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   4. expm1-define 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\color{blue}{\mathsf{expm1}\left(-0.5 \cdot \left(f \cdot \pi\right)\right)}}\right)}{\pi} \]

   5. associate-*r* 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\left(-0.5 \cdot f\right) \cdot \pi}\right)}\right)}{\pi} \]

   6. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\pi \cdot \left(-0.5 \cdot f\right)}\right)}\right)}{\pi} \]

   7. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \color{blue}{\left(f \cdot -0.5\right)}\right)}\right)}{\pi} \]

7. Simplified 98.9%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}} \]
8. Step-by-step derivation

   1. log1p-expm1-u 98.9%

      \[\leadsto -4 \cdot \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\right)\right)}}{\pi} \]

   2. expm1-undefine 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)} - 1}\right)}{\pi} \]

   3. add-exp-log 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)} - 1\right)}{\pi} \]

   4. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \color{blue}{\left(-0.5 \cdot f\right)}\right)}\right) - 1\right)}{\pi} \]

   5. associate-*r* 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\left(\pi \cdot -0.5\right) \cdot f}\right)}\right) - 1\right)}{\pi} \]

   6. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{f \cdot \left(\pi \cdot -0.5\right)}\right)}\right) - 1\right)}{\pi} \]

9. Applied egg-rr 98.9%

   \[\leadsto -4 \cdot \frac{\color{blue}{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right) - 1\right)}}{\pi} \]
10. Step-by-step derivation

    1. sub-neg 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right) + \left(-1\right)}\right)}{\pi} \]

    2. sub-neg 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \left(-\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right)\right)} + \left(-1\right)\right)}{\pi} \]

    3. distribute-neg-frac 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \color{blue}{\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}}\right) + \left(-1\right)\right)}{\pi} \]

    4. metadata-eval 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{\color{blue}{-1}}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right) + \left(-1\right)\right)}{\pi} \]

    5. metadata-eval 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right) + \color{blue}{-1}\right)}{\pi} \]

    6. associate-+l+ 99.0%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} + -1\right)}\right)}{\pi} \]

11. Simplified 99.0%

    \[\leadsto -4 \cdot \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} + -1\right)\right)}}{\pi} \]

12. Taylor expanded in f around 0 96.7%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\frac{f \cdot \left(f \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right) - 1\right) + 4 \cdot \frac{1}{\pi}}{f}}\right)}{\pi} \]
13. Step-by-step derivation

    1. pow1 96.7%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(\color{blue}{{\left(f \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)}^{1}} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

    2. distribute-rgt-out 96.7%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left({\left(f \cdot \left(\color{blue}{\pi \cdot \left(-0.08333333333333333 + 0.125\right)} - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)}^{1} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

    3. metadata-eval 96.7%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left({\left(f \cdot \left(\pi \cdot \color{blue}{0.041666666666666664} - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)}^{1} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

    4. distribute-rgt-out 96.7%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left({\left(f \cdot \left(\pi \cdot 0.041666666666666664 - \color{blue}{\pi \cdot \left(-0.125 + 0.08333333333333333\right)}\right)\right)}^{1} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

    5. metadata-eval 96.7%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left({\left(f \cdot \left(\pi \cdot 0.041666666666666664 - \pi \cdot \color{blue}{-0.041666666666666664}\right)\right)}^{1} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

14. Applied egg-rr 96.7%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(\color{blue}{{\left(f \cdot \left(\pi \cdot 0.041666666666666664 - \pi \cdot -0.041666666666666664\right)\right)}^{1}} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]
15. Step-by-step derivation

    1. unpow1 96.7%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(\color{blue}{f \cdot \left(\pi \cdot 0.041666666666666664 - \pi \cdot -0.041666666666666664\right)} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

    2. distribute-lft-out-- 96.7%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(f \cdot \color{blue}{\left(\pi \cdot \left(0.041666666666666664 - -0.041666666666666664\right)\right)} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

    3. metadata-eval 96.7%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(f \cdot \left(\pi \cdot \color{blue}{0.08333333333333333}\right) - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

    4. *-commutative 96.7%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(\color{blue}{\left(\pi \cdot 0.08333333333333333\right) \cdot f} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

    5. associate-*l* 96.7%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(\color{blue}{\pi \cdot \left(0.08333333333333333 \cdot f\right)} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

16. Simplified 96.7%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(\color{blue}{\pi \cdot \left(0.08333333333333333 \cdot f\right)} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

17. Final simplification 96.7%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(-1 + \pi \cdot \left(f \cdot 0.08333333333333333\right)\right) + \frac{1}{\pi} \cdot 4}{f}\right)}{\pi} \]
18. Add Preprocessing

Alternative 6: 96.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{4}{f \cdot \pi}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* -4.0 (/ (log (/ 4.0 (* f PI))) PI)))

C

double code(double f) {
	return -4.0 * (log((4.0 / (f * ((double) M_PI)))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log((4.0 / (f * Math.PI))) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log((4.0 / (f * math.pi))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log(Float64(4.0 / Float64(f * pi))) / pi))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 * (log((4.0 / (f * pi))) / pi);
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(4.0 / N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

3. Simplified 98.8%

   \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around inf 6.8%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{e^{0.5 \cdot \left(f \cdot \pi\right)} - 1} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi}} \]
6. Step-by-step derivation

   1. expm1-define 7.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\color{blue}{\mathsf{expm1}\left(0.5 \cdot \left(f \cdot \pi\right)\right)}} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   2. *-commutative 7.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\left(f \cdot \pi\right) \cdot 0.5}\right)} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   3. associate-*l* 7.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\color{blue}{f \cdot \left(\pi \cdot 0.5\right)}\right)} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   4. expm1-define 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\color{blue}{\mathsf{expm1}\left(-0.5 \cdot \left(f \cdot \pi\right)\right)}}\right)}{\pi} \]

   5. associate-*r* 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\left(-0.5 \cdot f\right) \cdot \pi}\right)}\right)}{\pi} \]

   6. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\pi \cdot \left(-0.5 \cdot f\right)}\right)}\right)}{\pi} \]

   7. *-commutative 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \color{blue}{\left(f \cdot -0.5\right)}\right)}\right)}{\pi} \]

7. Simplified 98.9%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}} \]

8. Taylor expanded in f around 0 96.0%

   \[\leadsto -4 \cdot \frac{\log \color{blue}{\left(\frac{4}{f \cdot \pi}\right)}}{\pi} \]
9. Step-by-step derivation

   1. *-commutative 96.0%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{4}{\color{blue}{\pi \cdot f}}\right)}{\pi} \]

10. Simplified 96.0%

    \[\leadsto -4 \cdot \frac{\log \color{blue}{\left(\frac{4}{\pi \cdot f}\right)}}{\pi} \]

11. Final simplification 96.0%

    \[\leadsto -4 \cdot \frac{\log \left(\frac{4}{f \cdot \pi}\right)}{\pi} \]
12. Add Preprocessing

Alternative 7: 5.0% accurate, 532.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (f) :precision binary64 0.0)

C

double code(double f) {
	return 0.0;
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = 0.0d0
end function

Java

public static double code(double f) {
	return 0.0;
}

Python

def code(f):
	return 0.0

Julia

function code(f)
	return 0.0
end

MATLAB

function tmp = code(f)
	tmp = 0.0;
end

Wolfram

code[f_] := 0.0

Derivation

1. Initial program 7.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

3. Simplified 98.8%

   \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

6. Applied egg-rr 94.3%

   \[\leadsto \color{blue}{\sqrt[3]{{\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right)}^{3}}} \cdot \frac{-4}{\pi} \]
7. Step-by-step derivation

   1. rem-cbrt-cube 94.7%

      \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right)} \cdot \frac{-4}{\pi} \]

   2. flip-+ 0.0%

      \[\leadsto \log \color{blue}{\left(\frac{\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} \cdot \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} - \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} \cdot \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}}{\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} - \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}}\right)} \cdot \frac{-4}{\pi} \]

   3. log-div 0.0%

      \[\leadsto \color{blue}{\left(\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} \cdot \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} - \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} \cdot \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right) - \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} - \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right)\right)} \cdot \frac{-4}{\pi} \]

8. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{\left(\log \left({\left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right)}^{2} - {\left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right)}^{2}\right) - \log \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} - \frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right)\right)} \cdot \frac{-4}{\pi} \]
9. Step-by-step derivation

   1. +-inverses 0.0%

      \[\leadsto \left(\log \color{blue}{0} - \log \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} - \frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right)\right) \cdot \frac{-4}{\pi} \]

   2. +-inverses 0.0%

      \[\leadsto \left(\log 0 - \log \color{blue}{0}\right) \cdot \frac{-4}{\pi} \]

   3. +-inverses 4.6%

      \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]

10. Simplified 4.6%

    \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]
11. Step-by-step derivation

    1. mul0-lft 4.6%

       \[\leadsto \color{blue}{0} \]

12. Applied egg-rr 4.6%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  :precision binary64
  (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))) (- (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))))))))