VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 99.0%

Time: 20.5s

Alternatives: 10

Speedup: 4.7×

• Rival profile vector overflowed, profile may not be complete

• could not determine a ground truth

  1. f = 2.00000000000000007e154

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: 6.8% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

3. Simplified 98.9%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 0.7%

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

   3. swap-sqr 0.7%

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

   4. metadata-eval 0.7%

      \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\sqrt{\color{blue}{0.25} \cdot \left(\pi \cdot \pi\right)} \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi} \]

   5. metadata-eval 0.7%

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

   6. swap-sqr 0.7%

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

   7. sqrt-unprod 0.2%

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

   8. add-sqr-sqrt 0.7%

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

   9. add-cube-cbrt 4.3%

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

   10. pow3 4.3%

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

6. Applied egg-rr 98.9%

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

   1. add-cbrt-cube 98.6%

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

   2. pow3 98.6%

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

8. Applied egg-rr 98.6%

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

   1. rem-cbrt-cube 98.9%

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

   2. log1p-expm1-u 98.9%

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

   3. log1p-undefine 98.9%

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

   4. expm1-undefine 98.9%

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

   5. add-exp-log 98.9%

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

   6. *-commutative 98.9%

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

   7. *-commutative 98.9%

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

   8. associate-*l* 98.9%

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

10. Applied egg-rr 98.9%

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

    1. log1p-define 98.9%

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

    2. associate--l+ 99.0%

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

    3. sub-neg 99.0%

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

    4. *-commutative 99.0%

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

    5. metadata-eval 99.0%

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

12. Simplified 99.0%

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

    1. add-sqr-sqrt 99.0%

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

    2. pow2 99.0%

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

14. Applied egg-rr 99.0%

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

15. Final simplification 99.0%

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

Alternative 2: 98.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 225.0)
   (-
    (* -4.0 (/ (- (log (/ 4.0 PI)) (log f)) PI))
    (* (pow f 2.0) (* PI 0.08333333333333333)))
   0.0))

C

double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = (-4.0 * ((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI))) - (pow(f, 2.0) * (((double) M_PI) * 0.08333333333333333));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = (-4.0 * ((Math.log((4.0 / Math.PI)) - Math.log(f)) / Math.PI)) - (Math.pow(f, 2.0) * (Math.PI * 0.08333333333333333));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 225.0:
		tmp = (-4.0 * ((math.log((4.0 / math.pi)) - math.log(f)) / math.pi)) - (math.pow(f, 2.0) * (math.pi * 0.08333333333333333))
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 225.0)
		tmp = Float64(Float64(-4.0 * Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi)) - Float64((f ^ 2.0) * Float64(pi * 0.08333333333333333)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 225.0)
		tmp = (-4.0 * ((log((4.0 / pi)) - log(f)) / pi)) - ((f ^ 2.0) * (pi * 0.08333333333333333));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 225.0], N[(N[(-4.0 * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] - N[(N[Power[f, 2.0], $MachinePrecision] * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 225

   1. Initial program 6.7%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in f around 0 98.6%

      \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}{\pi} + -1 \cdot \left({f}^{2} \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.6%

         \[\leadsto -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}{\pi} + \color{blue}{\left(-{f}^{2} \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)} \]

      2. unsub-neg 98.6%

         \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}{\pi} - {f}^{2} \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)} \]

      3. mul-1-neg 98.6%

         \[\leadsto -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(-\log f\right)}}{\pi} - {f}^{2} \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right) \]

      4. unsub-neg 98.6%

         \[\leadsto -4 \cdot \frac{\color{blue}{\log \left(\frac{4}{\pi}\right) - \log f}}{\pi} - {f}^{2} \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right) \]

      5. distribute-rgt-out 98.6%

         \[\leadsto -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} - {f}^{2} \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \color{blue}{\pi \cdot \left(-0.125 + 0.08333333333333333\right)}\right) \]

      6. metadata-eval 98.6%

         \[\leadsto -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} - {f}^{2} \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \pi \cdot \color{blue}{-0.041666666666666664}\right) \]

   7. Simplified 98.6%

      \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right)} \]

   if 225 < f

   1. Initial program 0.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 3.1%

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

      1. flip-+ 0.0%

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

      2. log-div 0.0%

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

   8. Applied egg-rr 0.0%

      \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{\left(\log \left(\frac{1}{{\left(\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)\right)}^{2}} - \frac{1}{{\left(\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)\right)}^{2}}\right) - \log \left(\frac{0}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right)\right)} \cdot -4}} \]
   9. Step-by-step derivation

      1. +-inverses 0.0%

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

      2. div0 0.0%

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

      3. +-inverses 100.0%

         \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]
   11. Step-by-step derivation

       1. metadata-eval 100.0%

          \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0}}} \]

       2. clear-num 100.0%

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

       3. div0 100.0%

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

   12. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

3. Simplified 98.9%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 0.7%

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

   3. swap-sqr 0.7%

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

   4. metadata-eval 0.7%

      \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\sqrt{\color{blue}{0.25} \cdot \left(\pi \cdot \pi\right)} \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi} \]

   5. metadata-eval 0.7%

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

   6. swap-sqr 0.7%

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

   7. sqrt-unprod 0.2%

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

   8. add-sqr-sqrt 0.7%

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

   9. add-cube-cbrt 4.3%

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

   10. pow3 4.3%

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

6. Applied egg-rr 98.9%

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

   1. add-cbrt-cube 98.6%

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

   2. pow3 98.6%

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

8. Applied egg-rr 98.6%

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

   1. rem-cbrt-cube 98.9%

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

   2. log1p-expm1-u 98.9%

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

   3. log1p-undefine 98.9%

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

   4. expm1-undefine 98.9%

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

   5. add-exp-log 98.9%

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

   6. *-commutative 98.9%

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

   7. *-commutative 98.9%

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

   8. associate-*l* 98.9%

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

10. Applied egg-rr 98.9%

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

    1. log1p-define 98.9%

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

    2. associate--l+ 99.0%

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

    3. sub-neg 99.0%

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

    4. *-commutative 99.0%

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

    5. metadata-eval 99.0%

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

12. Simplified 99.0%

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

13. Final simplification 99.0%

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

Alternative 4: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 5: 98.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{f \cdot \left(0.5 + f \cdot \left(\pi \cdot -0.08333333333333333 + \pi \cdot 0.125\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 225.0)
   (*
    (/ -4.0 PI)
    (log
     (+
      (/ 1.0 (expm1 (* f (* PI 0.5))))
      (/
       (+
        (* f (+ 0.5 (* f (+ (* PI -0.08333333333333333) (* PI 0.125)))))
        (* 2.0 (/ 1.0 PI)))
       f))))
   0.0))

C

double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = (-4.0 / ((double) M_PI)) * log(((1.0 / expm1((f * (((double) M_PI) * 0.5)))) + (((f * (0.5 + (f * ((((double) M_PI) * -0.08333333333333333) + (((double) M_PI) * 0.125))))) + (2.0 * (1.0 / ((double) M_PI)))) / f)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = (-4.0 / Math.PI) * Math.log(((1.0 / Math.expm1((f * (Math.PI * 0.5)))) + (((f * (0.5 + (f * ((Math.PI * -0.08333333333333333) + (Math.PI * 0.125))))) + (2.0 * (1.0 / Math.PI))) / f)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 225.0:
		tmp = (-4.0 / math.pi) * math.log(((1.0 / math.expm1((f * (math.pi * 0.5)))) + (((f * (0.5 + (f * ((math.pi * -0.08333333333333333) + (math.pi * 0.125))))) + (2.0 * (1.0 / math.pi))) / f)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 225.0)
		tmp = Float64(Float64(-4.0 / pi) * log(Float64(Float64(1.0 / expm1(Float64(f * Float64(pi * 0.5)))) + Float64(Float64(Float64(f * Float64(0.5 + Float64(f * Float64(Float64(pi * -0.08333333333333333) + Float64(pi * 0.125))))) + Float64(2.0 * Float64(1.0 / pi))) / f))));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[f_] := If[LessEqual[f, 225.0], N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(N[(1.0 / N[(Exp[N[(f * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(f * N[(0.5 + N[(f * N[(N[(Pi * -0.08333333333333333), $MachinePrecision] + N[(Pi * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 225

   1. Initial program 6.7%

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

   3. Simplified 98.9%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in f around 0 98.5%

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

   if 225 < f

   1. Initial program 0.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 3.1%

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

      1. flip-+ 0.0%

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

      2. log-div 0.0%

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

   8. Applied egg-rr 0.0%

      \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{\left(\log \left(\frac{1}{{\left(\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)\right)}^{2}} - \frac{1}{{\left(\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)\right)}^{2}}\right) - \log \left(\frac{0}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right)\right)} \cdot -4}} \]
   9. Step-by-step derivation

      1. +-inverses 0.0%

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

      2. div0 0.0%

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

      3. +-inverses 100.0%

         \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]
   11. Step-by-step derivation

       1. metadata-eval 100.0%

          \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0}}} \]

       2. clear-num 100.0%

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

       3. div0 100.0%

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

   12. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{f \cdot \left(0.5 + f \cdot \left(\pi \cdot -0.08333333333333333 + \pi \cdot 0.125\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{\left(\frac{2}{\pi} + {f}^{2} \cdot \left(\left(\pi \cdot -0.5\right) \cdot -0.08333333333333333 - \pi \cdot -0.041666666666666664\right)\right) + \frac{-1}{\pi \cdot -0.5}}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 225.0)
   (*
    (/ -4.0 PI)
    (log
     (/
      (+
       (+
        (/ 2.0 PI)
        (*
         (pow f 2.0)
         (-
          (* (* PI -0.5) -0.08333333333333333)
          (* PI -0.041666666666666664))))
       (/ -1.0 (* PI -0.5)))
      f)))
   0.0))

C

double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = (-4.0 / ((double) M_PI)) * log(((((2.0 / ((double) M_PI)) + (pow(f, 2.0) * (((((double) M_PI) * -0.5) * -0.08333333333333333) - (((double) M_PI) * -0.041666666666666664)))) + (-1.0 / (((double) M_PI) * -0.5))) / f));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = (-4.0 / Math.PI) * Math.log(((((2.0 / Math.PI) + (Math.pow(f, 2.0) * (((Math.PI * -0.5) * -0.08333333333333333) - (Math.PI * -0.041666666666666664)))) + (-1.0 / (Math.PI * -0.5))) / f));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 225.0:
		tmp = (-4.0 / math.pi) * math.log(((((2.0 / math.pi) + (math.pow(f, 2.0) * (((math.pi * -0.5) * -0.08333333333333333) - (math.pi * -0.041666666666666664)))) + (-1.0 / (math.pi * -0.5))) / f))
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 225.0)
		tmp = Float64(Float64(-4.0 / pi) * log(Float64(Float64(Float64(Float64(2.0 / pi) + Float64((f ^ 2.0) * Float64(Float64(Float64(pi * -0.5) * -0.08333333333333333) - Float64(pi * -0.041666666666666664)))) + Float64(-1.0 / Float64(pi * -0.5))) / f)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 225.0)
		tmp = (-4.0 / pi) * log(((((2.0 / pi) + ((f ^ 2.0) * (((pi * -0.5) * -0.08333333333333333) - (pi * -0.041666666666666664)))) + (-1.0 / (pi * -0.5))) / f));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 225.0], N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(N[(N[(N[(2.0 / Pi), $MachinePrecision] + N[(N[Power[f, 2.0], $MachinePrecision] * N[(N[(N[(Pi * -0.5), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] - N[(Pi * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(Pi * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 225

   1. Initial program 6.7%

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

   3. Simplified 98.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.6%

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

      3. swap-sqr 0.6%

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

      4. metadata-eval 0.6%

         \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\sqrt{\color{blue}{0.25} \cdot \left(\pi \cdot \pi\right)} \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi} \]

      5. metadata-eval 0.6%

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

      6. swap-sqr 0.6%

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

      7. sqrt-unprod 0.2%

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

      8. add-sqr-sqrt 0.6%

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

      9. add-cube-cbrt 4.3%

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

      10. pow3 4.4%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in f around 0 98.5%

      \[\leadsto \log \color{blue}{\left(\frac{\left(2 \cdot \frac{1}{\pi} + {f}^{2} \cdot \left(\left(-0.25 \cdot \left(\pi \cdot {\left(\sqrt[3]{-0.5}\right)}^{3}\right) + 0.16666666666666666 \cdot \left(\pi \cdot {\left(\sqrt[3]{-0.5}\right)}^{3}\right)\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) - \frac{1}{\pi \cdot {\left(\sqrt[3]{-0.5}\right)}^{3}}}{f}\right)} \cdot \frac{-4}{\pi} \]
   8. Step-by-step derivation

   9. Simplified 98.5%

      \[\leadsto \log \color{blue}{\left(\frac{\left(\frac{2}{\pi} + {f}^{2} \cdot \left(\left(\pi \cdot -0.5\right) \cdot -0.08333333333333333 - \pi \cdot -0.041666666666666664\right)\right) - \frac{1}{\pi \cdot -0.5}}{f}\right)} \cdot \frac{-4}{\pi} \]

   if 225 < f

   1. Initial program 0.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 3.1%

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

      1. flip-+ 0.0%

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

      2. log-div 0.0%

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

   8. Applied egg-rr 0.0%

      \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{\left(\log \left(\frac{1}{{\left(\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)\right)}^{2}} - \frac{1}{{\left(\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)\right)}^{2}}\right) - \log \left(\frac{0}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right)\right)} \cdot -4}} \]
   9. Step-by-step derivation

      1. +-inverses 0.0%

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

      2. div0 0.0%

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

      3. +-inverses 100.0%

         \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]
   11. Step-by-step derivation

       1. metadata-eval 100.0%

          \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0}}} \]

       2. clear-num 100.0%

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

       3. div0 100.0%

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

   12. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{\left(\frac{2}{\pi} + {f}^{2} \cdot \left(\left(\pi \cdot -0.5\right) \cdot -0.08333333333333333 - \pi \cdot -0.041666666666666664\right)\right) + \frac{-1}{\pi \cdot -0.5}}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;\frac{-4}{\pi} \cdot \mathsf{log1p}\left(\frac{f \cdot \left(-1 + f \cdot \left(\left(\pi \cdot -0.08333333333333333 + \pi \cdot 0.125\right) - \left(\pi \cdot 0.08333333333333333 + \pi \cdot -0.125\right)\right)\right) + 4 \cdot \frac{1}{\pi}}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 225.0)
   (*
    (/ -4.0 PI)
    (log1p
     (/
      (+
       (*
        f
        (+
         -1.0
         (*
          f
          (-
           (+ (* PI -0.08333333333333333) (* PI 0.125))
           (+ (* PI 0.08333333333333333) (* PI -0.125))))))
       (* 4.0 (/ 1.0 PI)))
      f)))
   0.0))

C

double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = (-4.0 / ((double) M_PI)) * log1p((((f * (-1.0 + (f * (((((double) M_PI) * -0.08333333333333333) + (((double) M_PI) * 0.125)) - ((((double) M_PI) * 0.08333333333333333) + (((double) M_PI) * -0.125)))))) + (4.0 * (1.0 / ((double) M_PI)))) / f));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = (-4.0 / Math.PI) * Math.log1p((((f * (-1.0 + (f * (((Math.PI * -0.08333333333333333) + (Math.PI * 0.125)) - ((Math.PI * 0.08333333333333333) + (Math.PI * -0.125)))))) + (4.0 * (1.0 / Math.PI))) / f));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 225.0:
		tmp = (-4.0 / math.pi) * math.log1p((((f * (-1.0 + (f * (((math.pi * -0.08333333333333333) + (math.pi * 0.125)) - ((math.pi * 0.08333333333333333) + (math.pi * -0.125)))))) + (4.0 * (1.0 / math.pi))) / f))
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 225.0)
		tmp = Float64(Float64(-4.0 / pi) * log1p(Float64(Float64(Float64(f * Float64(-1.0 + Float64(f * Float64(Float64(Float64(pi * -0.08333333333333333) + Float64(pi * 0.125)) - Float64(Float64(pi * 0.08333333333333333) + Float64(pi * -0.125)))))) + Float64(4.0 * Float64(1.0 / pi))) / f)));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[f_] := If[LessEqual[f, 225.0], N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[1 + N[(N[(N[(f * N[(-1.0 + N[(f * N[(N[(N[(Pi * -0.08333333333333333), $MachinePrecision] + N[(Pi * 0.125), $MachinePrecision]), $MachinePrecision] - N[(N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(Pi * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 225

   1. Initial program 6.7%

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

   3. Simplified 98.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 0.6%

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

      3. swap-sqr 0.6%

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

      4. metadata-eval 0.6%

         \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\sqrt{\color{blue}{0.25} \cdot \left(\pi \cdot \pi\right)} \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi} \]

      5. metadata-eval 0.6%

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

      6. swap-sqr 0.6%

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

      7. sqrt-unprod 0.2%

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

      8. add-sqr-sqrt 0.6%

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

      9. add-cube-cbrt 4.3%

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

      10. pow3 4.4%

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

   6. Applied egg-rr 98.9%

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

      1. add-cbrt-cube 98.6%

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

      2. pow3 98.6%

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

   8. Applied egg-rr 98.6%

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

      1. rem-cbrt-cube 98.9%

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

      2. log1p-expm1-u 98.9%

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

      3. log1p-undefine 98.9%

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

      4. expm1-undefine 98.9%

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

      5. add-exp-log 98.9%

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

      6. *-commutative 98.9%

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

      7. *-commutative 98.9%

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

      8. associate-*l* 98.9%

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

   10. Applied egg-rr 98.9%

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

       1. log1p-define 98.9%

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

       2. associate--l+ 99.0%

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

       3. sub-neg 99.0%

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

       4. *-commutative 99.0%

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

       5. metadata-eval 99.0%

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

   12. Simplified 99.0%

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

   13. Taylor expanded in f around 0 98.5%

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

   if 225 < f

   1. Initial program 0.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 3.1%

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

      1. flip-+ 0.0%

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

      2. log-div 0.0%

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

   8. Applied egg-rr 0.0%

      \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{\left(\log \left(\frac{1}{{\left(\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)\right)}^{2}} - \frac{1}{{\left(\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)\right)}^{2}}\right) - \log \left(\frac{0}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right)\right)} \cdot -4}} \]
   9. Step-by-step derivation

      1. +-inverses 0.0%

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

      2. div0 0.0%

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

      3. +-inverses 100.0%

         \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]
   11. Step-by-step derivation

       1. metadata-eval 100.0%

          \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0}}} \]

       2. clear-num 100.0%

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

       3. div0 100.0%

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

   12. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;\frac{-4}{\pi} \cdot \mathsf{log1p}\left(\frac{f \cdot \left(-1 + f \cdot \left(\left(\pi \cdot -0.08333333333333333 + \pi \cdot 0.125\right) - \left(\pi \cdot 0.08333333333333333 + \pi \cdot -0.125\right)\right)\right) + 4 \cdot \frac{1}{\pi}}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.25:\\ \;\;\;\;\frac{\log \left(\frac{4}{f \cdot \pi}\right)}{\pi \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.25) (/ (log (/ 4.0 (* f PI))) (* PI -0.25)) 0.0))

C

double code(double f) {
	double tmp;
	if (f <= 1.25) {
		tmp = log((4.0 / (f * ((double) M_PI)))) / (((double) M_PI) * -0.25);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.25) {
		tmp = Math.log((4.0 / (f * Math.PI))) / (Math.PI * -0.25);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.25:
		tmp = math.log((4.0 / (f * math.pi))) / (math.pi * -0.25)
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.25)
		tmp = Float64(log(Float64(4.0 / Float64(f * pi))) / Float64(pi * -0.25));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.25)
		tmp = log((4.0 / (f * pi))) / (pi * -0.25);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.25], N[(N[Log[N[(4.0 / N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * -0.25), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 1.25

   1. Initial program 6.7%

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

   3. Simplified 99.3%

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

   6. Applied egg-rr 97.6%

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

   7. Taylor expanded in f around inf 2.9%

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

      1. associate-*r/ 2.9%

         \[\leadsto \frac{1}{\frac{\pi}{\log \left(-\color{blue}{\frac{2 \cdot 1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}}\right) \cdot -4}} \]

      2. metadata-eval 2.9%

         \[\leadsto \frac{1}{\frac{\pi}{\log \left(-\frac{\color{blue}{2}}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right) \cdot -4}} \]

      3. expm1-define 97.6%

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

      4. associate-*r* 97.6%

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

      5. *-commutative 97.6%

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

      6. distribute-neg-frac 97.6%

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

      7. metadata-eval 97.6%

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

   9. Simplified 97.6%

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

   10. Taylor expanded in f around 0 98.6%

       \[\leadsto \frac{1}{\color{blue}{-0.25 \cdot \frac{\pi}{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 98.6%

          \[\leadsto \frac{1}{\color{blue}{\frac{-0.25 \cdot \pi}{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}}} \]

       2. *-commutative 98.6%

          \[\leadsto \frac{1}{\frac{\color{blue}{\pi \cdot -0.25}}{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}} \]

       3. mul-1-neg 98.6%

          \[\leadsto \frac{1}{\frac{\pi \cdot -0.25}{\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(-\log f\right)}}} \]

       4. unsub-neg 98.6%

          \[\leadsto \frac{1}{\frac{\pi \cdot -0.25}{\color{blue}{\log \left(\frac{4}{\pi}\right) - \log f}}} \]

   12. Simplified 98.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{\pi \cdot -0.25}{\log \left(\frac{4}{\pi}\right) - \log f}}} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 98.6%

          \[\leadsto \color{blue}{1 \cdot \frac{1}{\frac{\pi \cdot -0.25}{\log \left(\frac{4}{\pi}\right) - \log f}}} \]

       2. associate-/r/ 98.6%

          \[\leadsto 1 \cdot \color{blue}{\left(\frac{1}{\pi \cdot -0.25} \cdot \left(\log \left(\frac{4}{\pi}\right) - \log f\right)\right)} \]

       3. diff-log 98.6%

          \[\leadsto 1 \cdot \left(\frac{1}{\pi \cdot -0.25} \cdot \color{blue}{\log \left(\frac{\frac{4}{\pi}}{f}\right)}\right) \]

   14. Applied egg-rr 98.6%

       \[\leadsto \color{blue}{1 \cdot \left(\frac{1}{\pi \cdot -0.25} \cdot \log \left(\frac{\frac{4}{\pi}}{f}\right)\right)} \]
   15. Step-by-step derivation

       1. *-lft-identity 98.6%

          \[\leadsto \color{blue}{\frac{1}{\pi \cdot -0.25} \cdot \log \left(\frac{\frac{4}{\pi}}{f}\right)} \]

       2. associate-*l/ 98.8%

          \[\leadsto \color{blue}{\frac{1 \cdot \log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi \cdot -0.25}} \]

       3. *-lft-identity 98.8%

          \[\leadsto \frac{\color{blue}{\log \left(\frac{\frac{4}{\pi}}{f}\right)}}{\pi \cdot -0.25} \]

       4. associate-/l/ 98.8%

          \[\leadsto \frac{\log \color{blue}{\left(\frac{4}{f \cdot \pi}\right)}}{\pi \cdot -0.25} \]

       5. *-commutative 98.8%

          \[\leadsto \frac{\log \left(\frac{4}{\color{blue}{\pi \cdot f}}\right)}{\pi \cdot -0.25} \]

   16. Simplified 98.8%

       \[\leadsto \color{blue}{\frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi \cdot -0.25}} \]

   if 1.25 < f

   1. Initial program 0.6%

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

   3. Simplified 83.9%

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

   6. Applied egg-rr 3.9%

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

      1. flip-+ 0.0%

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

      2. log-div 0.0%

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

   8. Applied egg-rr 0.0%

      \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{\left(\log \left(\frac{1}{{\left(\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)\right)}^{2}} - \frac{1}{{\left(\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)\right)}^{2}}\right) - \log \left(\frac{0}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right)\right)} \cdot -4}} \]
   9. Step-by-step derivation

      1. +-inverses 0.0%

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

      2. div0 0.0%

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

      3. +-inverses 83.9%

         \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]

   10. Simplified 83.9%

       \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]
   11. Step-by-step derivation

       1. metadata-eval 83.9%

          \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0}}} \]

       2. clear-num 83.9%

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

       3. div0 83.9%

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

   12. Applied egg-rr 83.9%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.25:\\ \;\;\;\;\frac{\log \left(\frac{4}{f \cdot \pi}\right)}{\pi \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.25:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{4}{f \cdot \pi}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.25) (* (/ -4.0 PI) (log (/ 4.0 (* f PI)))) 0.0))

C

double code(double f) {
	double tmp;
	if (f <= 1.25) {
		tmp = (-4.0 / ((double) M_PI)) * log((4.0 / (f * ((double) M_PI))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.25) {
		tmp = (-4.0 / Math.PI) * Math.log((4.0 / (f * Math.PI)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.25:
		tmp = (-4.0 / math.pi) * math.log((4.0 / (f * math.pi)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.25)
		tmp = Float64(Float64(-4.0 / pi) * log(Float64(4.0 / Float64(f * pi))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.25)
		tmp = (-4.0 / pi) * log((4.0 / (f * pi)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.25], N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(4.0 / N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 1.25

   1. Initial program 6.7%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in f around 0 98.6%

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

      1. *-commutative 98.6%

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

   7. Simplified 98.6%

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

   if 1.25 < f

   1. Initial program 0.6%

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

   3. Simplified 83.9%

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

   6. Applied egg-rr 3.9%

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

      1. flip-+ 0.0%

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

      2. log-div 0.0%

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

   8. Applied egg-rr 0.0%

      \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{\left(\log \left(\frac{1}{{\left(\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)\right)}^{2}} - \frac{1}{{\left(\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)\right)}^{2}}\right) - \log \left(\frac{0}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right)\right)} \cdot -4}} \]
   9. Step-by-step derivation

      1. +-inverses 0.0%

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

      2. div0 0.0%

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

      3. +-inverses 83.9%

         \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]

   10. Simplified 83.9%

       \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]
   11. Step-by-step derivation

       1. metadata-eval 83.9%

          \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0}}} \]

       2. clear-num 83.9%

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

       3. div0 83.9%

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

   12. Applied egg-rr 83.9%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.25:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{4}{f \cdot \pi}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 4.9% 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 6.6%

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

3. Simplified 98.9%

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

6. Applied egg-rr 95.4%

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

   1. flip-+ 0.0%

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

   2. log-div 0.0%

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

8. Applied egg-rr 0.0%

   \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{\left(\log \left(\frac{1}{{\left(\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)\right)}^{2}} - \frac{1}{{\left(\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)\right)}^{2}}\right) - \log \left(\frac{0}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right)\right)} \cdot -4}} \]
9. Step-by-step derivation

   1. +-inverses 0.0%

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

   2. div0 0.0%

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

   3. +-inverses 5.0%

      \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]

10. Simplified 5.0%

    \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]
11. Step-by-step derivation

    1. metadata-eval 5.0%

       \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0}}} \]

    2. clear-num 5.0%

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

    3. div0 5.0%

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

12. Applied egg-rr 5.0%

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

Reproduce

herbie shell --seed 2024156 
(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)))))))))