VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 96.5%

Time: 27.5s

Alternatives: 8

Speedup: 3.3×

• 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: 6.6% 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: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, -2 \cdot \frac{{\pi}^{3}}{{\pi}^{2} \cdot 48}\right), 0 \cdot {\left(\pi \cdot 0.5\right)}^{2}\right), -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (fma
  -2.0
  (+
   (* (/ f PI) (* PI 0.0))
   (*
    (/ (pow f 2.0) PI)
    (fma
     (* PI 0.5)
     (fma
      0.0625
      (/ (pow PI 2.0) (* PI 0.5))
      (* -2.0 (/ (pow PI 3.0) (* (pow PI 2.0) 48.0))))
     (* 0.0 (pow (* PI 0.5) 2.0)))))
  (* -4.0 (/ (- (log (/ 2.0 (* PI 0.5))) (log f)) PI))))

C

double code(double f) {
	return fma(-2.0, (((f / ((double) M_PI)) * (((double) M_PI) * 0.0)) + ((pow(f, 2.0) / ((double) M_PI)) * fma((((double) M_PI) * 0.5), fma(0.0625, (pow(((double) M_PI), 2.0) / (((double) M_PI) * 0.5)), (-2.0 * (pow(((double) M_PI), 3.0) / (pow(((double) M_PI), 2.0) * 48.0)))), (0.0 * pow((((double) M_PI) * 0.5), 2.0))))), (-4.0 * ((log((2.0 / (((double) M_PI) * 0.5))) - log(f)) / ((double) M_PI))));
}

Julia

function code(f)
	return fma(-2.0, Float64(Float64(Float64(f / pi) * Float64(pi * 0.0)) + Float64(Float64((f ^ 2.0) / pi) * fma(Float64(pi * 0.5), fma(0.0625, Float64((pi ^ 2.0) / Float64(pi * 0.5)), Float64(-2.0 * Float64((pi ^ 3.0) / Float64((pi ^ 2.0) * 48.0)))), Float64(0.0 * (Float64(pi * 0.5) ^ 2.0))))), Float64(-4.0 * Float64(Float64(log(Float64(2.0 / Float64(pi * 0.5))) - log(f)) / pi)))
end

Wolfram

code[f_] := N[(-2.0 * N[(N[(N[(f / Pi), $MachinePrecision] * N[(Pi * 0.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[f, 2.0], $MachinePrecision] / Pi), $MachinePrecision] * N[(N[(Pi * 0.5), $MachinePrecision] * N[(0.0625 * N[(N[Power[Pi, 2.0], $MachinePrecision] / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[Pi, 3.0], $MachinePrecision] / N[(N[Power[Pi, 2.0], $MachinePrecision] * 48.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0 * N[Power[N[(Pi * 0.5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(N[Log[N[(2.0 / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 6.9%

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

   2. *-commutative 6.9%

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

3. Simplified 6.9%

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

4. Taylor expanded in f around 0 97.3%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{f \cdot \left(\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}{\pi} + -2 \cdot \frac{{f}^{2} \cdot \left(-0.25 \cdot \left({\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2} \cdot {\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}\right) + \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right) \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}{\pi}\right)} \]

6. Simplified 97.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right), -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}\right)} \]
7. Step-by-step derivation

   1. expm1-log1p-u 97.3%

      \[\leadsto \mathsf{fma}\left(-2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}\right)\right)}} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right), -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}\right) \]

   2. expm1-udef 97.3%

      \[\leadsto \mathsf{fma}\left(-2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}\right)} - 1}} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right), -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}\right) \]

   3. div-inv 97.3%

      \[\leadsto \mathsf{fma}\left(-2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{e^{\mathsf{log1p}\left(\color{blue}{{\left(\pi \cdot 0.5\right)}^{2} \cdot \frac{1}{0.005208333333333333}}\right)} - 1} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right), -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}\right) \]

   4. *-commutative 97.3%

      \[\leadsto \mathsf{fma}\left(-2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{e^{\mathsf{log1p}\left({\color{blue}{\left(0.5 \cdot \pi\right)}}^{2} \cdot \frac{1}{0.005208333333333333}\right)} - 1} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right), -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}\right) \]

   5. unpow-prod-down 97.3%

      \[\leadsto \mathsf{fma}\left(-2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{e^{\mathsf{log1p}\left(\color{blue}{\left({0.5}^{2} \cdot {\pi}^{2}\right)} \cdot \frac{1}{0.005208333333333333}\right)} - 1} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right), -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}\right) \]

   6. metadata-eval 97.3%

      \[\leadsto \mathsf{fma}\left(-2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{e^{\mathsf{log1p}\left(\left(\color{blue}{0.25} \cdot {\pi}^{2}\right) \cdot \frac{1}{0.005208333333333333}\right)} - 1} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right), -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}\right) \]

   7. metadata-eval 97.3%

      \[\leadsto \mathsf{fma}\left(-2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{e^{\mathsf{log1p}\left(\left(0.25 \cdot {\pi}^{2}\right) \cdot \color{blue}{192}\right)} - 1} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right), -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}\right) \]

8. Applied egg-rr 97.3%

   \[\leadsto \mathsf{fma}\left(-2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\color{blue}{e^{\mathsf{log1p}\left(\left(0.25 \cdot {\pi}^{2}\right) \cdot 192\right)} - 1}} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right), -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}\right) \]
9. Step-by-step derivation

   1. expm1-def 97.3%

      \[\leadsto \mathsf{fma}\left(-2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.25 \cdot {\pi}^{2}\right) \cdot 192\right)\right)}} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right), -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}\right) \]

   2. expm1-log1p 97.3%

      \[\leadsto \mathsf{fma}\left(-2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\color{blue}{\left(0.25 \cdot {\pi}^{2}\right) \cdot 192}} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right), -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}\right) \]

   3. *-commutative 97.3%

      \[\leadsto \mathsf{fma}\left(-2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\color{blue}{\left({\pi}^{2} \cdot 0.25\right)} \cdot 192} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right), -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}\right) \]

   4. associate-*l* 97.3%

      \[\leadsto \mathsf{fma}\left(-2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\color{blue}{{\pi}^{2} \cdot \left(0.25 \cdot 192\right)}} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right), -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}\right) \]

   5. metadata-eval 97.3%

      \[\leadsto \mathsf{fma}\left(-2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{{\pi}^{2} \cdot \color{blue}{48}} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right), -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}\right) \]

10. Simplified 97.3%

    \[\leadsto \mathsf{fma}\left(-2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\color{blue}{{\pi}^{2} \cdot 48}} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right), -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}\right) \]

11. Final simplification 97.3%

    \[\leadsto \mathsf{fma}\left(-2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, -2 \cdot \frac{{\pi}^{3}}{{\pi}^{2} \cdot 48}\right), 0 \cdot {\left(\pi \cdot 0.5\right)}^{2}\right), -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}\right) \]

Alternative 2: 96.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}, -2 \cdot \left(\frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi, 0.125, \frac{{\pi}^{3} \cdot 0.005208333333333333}{{\pi}^{2}} \cdot -8\right)\right)\right) + 0 \cdot \frac{f}{2}\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (fma
  -4.0
  (/ (log (/ (/ 4.0 PI) f)) PI)
  (*
   -2.0
   (+
    (*
     (/ (pow f 2.0) PI)
     (*
      PI
      (*
       0.5
       (fma
        PI
        0.125
        (* (/ (* (pow PI 3.0) 0.005208333333333333) (pow PI 2.0)) -8.0)))))
    (* 0.0 (/ f 2.0))))))

C

double code(double f) {
	return fma(-4.0, (log(((4.0 / ((double) M_PI)) / f)) / ((double) M_PI)), (-2.0 * (((pow(f, 2.0) / ((double) M_PI)) * (((double) M_PI) * (0.5 * fma(((double) M_PI), 0.125, (((pow(((double) M_PI), 3.0) * 0.005208333333333333) / pow(((double) M_PI), 2.0)) * -8.0))))) + (0.0 * (f / 2.0)))));
}

Julia

function code(f)
	return fma(-4.0, Float64(log(Float64(Float64(4.0 / pi) / f)) / pi), Float64(-2.0 * Float64(Float64(Float64((f ^ 2.0) / pi) * Float64(pi * Float64(0.5 * fma(pi, 0.125, Float64(Float64(Float64((pi ^ 3.0) * 0.005208333333333333) / (pi ^ 2.0)) * -8.0))))) + Float64(0.0 * Float64(f / 2.0)))))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] + N[(-2.0 * N[(N[(N[(N[Power[f, 2.0], $MachinePrecision] / Pi), $MachinePrecision] * N[(Pi * N[(0.5 * N[(Pi * 0.125 + N[(N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0 * N[(f / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 6.9%

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

   2. *-commutative 6.9%

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

3. Simplified 6.9%

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

4. Taylor expanded in f around inf 6.9%

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

   1. associate-*r* 6.9%

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

   2. *-commutative 6.9%

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

   3. exp-prod 6.5%

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

   4. *-commutative 6.5%

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

   5. *-commutative 6.5%

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

   6. *-commutative 6.5%

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

   7. associate-*l* 6.5%

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

   8. exp-prod 6.5%

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

6. Simplified 6.5%

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

7. Taylor expanded in f around inf 6.9%

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

   1. exp-prod 6.5%

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

   2. *-commutative 6.5%

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

   3. associate-*r* 6.5%

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

   4. *-commutative 6.5%

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

   5. *-commutative 6.5%

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

   6. exp-prod 6.5%

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

9. Simplified 6.5%

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

10. Taylor expanded in f around 0 97.3%

    \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{f \cdot \left(\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}{\pi} + -2 \cdot \frac{{f}^{2} \cdot \left(-0.25 \cdot \left({\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2} \cdot {\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}\right) + \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right) \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}{\pi}\right)} \]

12. Simplified 97.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}, -2 \cdot \left(\frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.125 \cdot \frac{{\pi}^{2}}{\pi} + -2 \cdot \frac{{\pi}^{3} \cdot 0.005208333333333333}{0.25 \cdot {\pi}^{2}}, \left(0.25 \cdot {\pi}^{2}\right) \cdot 0\right) + \frac{f}{2} \cdot 0\right)\right)} \]
13. Step-by-step derivation

    1. fma-udef 97.3%

       \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}, -2 \cdot \left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\left(\pi \cdot 0.5\right) \cdot \left(0.125 \cdot \frac{{\pi}^{2}}{\pi} + -2 \cdot \frac{{\pi}^{3} \cdot 0.005208333333333333}{0.25 \cdot {\pi}^{2}}\right) + \left(0.25 \cdot {\pi}^{2}\right) \cdot 0\right)} + \frac{f}{2} \cdot 0\right)\right) \]

14. Applied egg-rr 97.3%

    \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}, -2 \cdot \left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, \frac{-2 \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)}{{\pi}^{2} \cdot 0.25}\right) + 0\right)} + \frac{f}{2} \cdot 0\right)\right) \]
15. Step-by-step derivation

    1. +-rgt-identity 97.3%

       \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}, -2 \cdot \left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, \frac{-2 \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)}{{\pi}^{2} \cdot 0.25}\right)\right)} + \frac{f}{2} \cdot 0\right)\right) \]

    2. associate-*l* 97.3%

       \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}, -2 \cdot \left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi, 0.125, \frac{-2 \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)}{{\pi}^{2} \cdot 0.25}\right)\right)\right)} + \frac{f}{2} \cdot 0\right)\right) \]

    3. *-commutative 97.3%

       \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}, -2 \cdot \left(\frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi, 0.125, \frac{\color{blue}{\left({\pi}^{3} \cdot 0.005208333333333333\right) \cdot -2}}{{\pi}^{2} \cdot 0.25}\right)\right)\right) + \frac{f}{2} \cdot 0\right)\right) \]

    4. times-frac 97.3%

       \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}, -2 \cdot \left(\frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi, 0.125, \color{blue}{\frac{{\pi}^{3} \cdot 0.005208333333333333}{{\pi}^{2}} \cdot \frac{-2}{0.25}}\right)\right)\right) + \frac{f}{2} \cdot 0\right)\right) \]

    5. metadata-eval 97.3%

       \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}, -2 \cdot \left(\frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi, 0.125, \frac{{\pi}^{3} \cdot 0.005208333333333333}{{\pi}^{2}} \cdot \color{blue}{-8}\right)\right)\right) + \frac{f}{2} \cdot 0\right)\right) \]

16. Simplified 97.3%

    \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}, -2 \cdot \left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi, 0.125, \frac{{\pi}^{3} \cdot 0.005208333333333333}{{\pi}^{2}} \cdot -8\right)\right)\right)} + \frac{f}{2} \cdot 0\right)\right) \]

17. Final simplification 97.3%

    \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}, -2 \cdot \left(\frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi, 0.125, \frac{{\pi}^{3} \cdot 0.005208333333333333}{{\pi}^{2}} \cdot -8\right)\right)\right) + 0 \cdot \frac{f}{2}\right)\right) \]

Alternative 3: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{f \cdot \frac{\pi}{4}}}{\mathsf{fma}\left(\pi, f \cdot 0.5, 0.005208333333333333 \cdot {\left(f \cdot \pi\right)}^{3}\right)}\right) \cdot \frac{-4}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    (+ (exp (/ PI (/ -4.0 f))) (exp (* f (/ PI 4.0))))
    (fma PI (* f 0.5) (* 0.005208333333333333 (pow (* f PI) 3.0)))))
  (/ -4.0 PI)))

C

double code(double f) {
	return log(((exp((((double) M_PI) / (-4.0 / f))) + exp((f * (((double) M_PI) / 4.0)))) / fma(((double) M_PI), (f * 0.5), (0.005208333333333333 * pow((f * ((double) M_PI)), 3.0))))) * (-4.0 / ((double) M_PI));
}

Julia

function code(f)
	return Float64(log(Float64(Float64(exp(Float64(pi / Float64(-4.0 / f))) + exp(Float64(f * Float64(pi / 4.0)))) / fma(pi, Float64(f * 0.5), Float64(0.005208333333333333 * (Float64(f * pi) ^ 3.0))))) * Float64(-4.0 / pi))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[Exp[N[(Pi / N[(-4.0 / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(f * N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Pi * N[(f * 0.5), $MachinePrecision] + N[(0.005208333333333333 * N[Power[N[(f * Pi), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 6.9%

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

   2. *-commutative 6.9%

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

3. Simplified 6.9%

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

4. Taylor expanded in f around inf 6.9%

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

   1. associate-*r* 6.9%

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

   2. *-commutative 6.9%

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

   3. exp-prod 6.5%

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

   4. *-commutative 6.5%

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

   5. *-commutative 6.5%

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

   6. *-commutative 6.5%

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

   7. associate-*l* 6.5%

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

   8. exp-prod 6.5%

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

6. Simplified 6.5%

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

7. Taylor expanded in f around inf 6.9%

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

   1. exp-prod 6.5%

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

   2. *-commutative 6.5%

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

   3. associate-*r* 6.5%

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

   4. *-commutative 6.5%

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

   5. *-commutative 6.5%

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

   6. exp-prod 6.5%

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

9. Simplified 6.5%

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

10. Taylor expanded in f around 0 97.1%

    \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}}\right) \cdot \frac{-4}{\pi} \]
11. Step-by-step derivation

    1. *-commutative 97.1%

       \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f} + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right) \cdot \frac{-4}{\pi} \]

    2. distribute-rgt-out-- 97.1%

       \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{\left(\pi \cdot \left(0.25 - -0.25\right)\right)} \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right) \cdot \frac{-4}{\pi} \]

    3. metadata-eval 97.1%

       \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\left(\pi \cdot \color{blue}{0.5}\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right) \cdot \frac{-4}{\pi} \]

    4. associate-*r* 97.1%

       \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{\pi \cdot \left(0.5 \cdot f\right)} + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right) \cdot \frac{-4}{\pi} \]

    5. fma-def 97.1%

       \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(\pi, 0.5 \cdot f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}}\right) \cdot \frac{-4}{\pi} \]

    6. *-commutative 97.1%

       \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(\pi, \color{blue}{f \cdot 0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

    7. distribute-rgt-out-- 97.1%

       \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(\pi, f \cdot 0.5, {f}^{3} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

    8. associate-*r* 97.1%

       \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(\pi, f \cdot 0.5, \color{blue}{\left({f}^{3} \cdot {\pi}^{3}\right) \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

    9. cube-prod 97.1%

       \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(\pi, f \cdot 0.5, \color{blue}{{\left(f \cdot \pi\right)}^{3}} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right) \cdot \frac{-4}{\pi} \]

    10. *-commutative 97.1%

        \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(\pi, f \cdot 0.5, {\color{blue}{\left(\pi \cdot f\right)}}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right) \cdot \frac{-4}{\pi} \]

    11. metadata-eval 97.1%

        \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(\pi, f \cdot 0.5, {\left(\pi \cdot f\right)}^{3} \cdot \color{blue}{0.005208333333333333}\right)}\right) \cdot \frac{-4}{\pi} \]

12. Simplified 97.1%

    \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(\pi, f \cdot 0.5, {\left(\pi \cdot f\right)}^{3} \cdot 0.005208333333333333\right)}}\right) \cdot \frac{-4}{\pi} \]

13. Final simplification 97.1%

    \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{f \cdot \frac{\pi}{4}}}{\mathsf{fma}\left(\pi, f \cdot 0.5, 0.005208333333333333 \cdot {\left(f \cdot \pi\right)}^{3}\right)}\right) \cdot \frac{-4}{\pi} \]

Alternative 4: 96.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} + -0.125 \cdot \left(\pi \cdot {f}^{2}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (+
  (* -4.0 (/ (- (log (/ 4.0 PI)) (log f)) PI))
  (* -0.125 (* PI (pow f 2.0)))))

C

double code(double f) {
	return (-4.0 * ((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI))) + (-0.125 * (((double) M_PI) * pow(f, 2.0)));
}

Java

public static double code(double f) {
	return (-4.0 * ((Math.log((4.0 / Math.PI)) - Math.log(f)) / Math.PI)) + (-0.125 * (Math.PI * Math.pow(f, 2.0)));
}

Python

def code(f):
	return (-4.0 * ((math.log((4.0 / math.pi)) - math.log(f)) / math.pi)) + (-0.125 * (math.pi * math.pow(f, 2.0)))

Julia

function code(f)
	return Float64(Float64(-4.0 * Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi)) + Float64(-0.125 * Float64(pi * (f ^ 2.0))))
end

MATLAB

function tmp = code(f)
	tmp = (-4.0 * ((log((4.0 / pi)) - log(f)) / pi)) + (-0.125 * (pi * (f ^ 2.0)));
end

Wolfram

code[f_] := N[(N[(-4.0 * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] + N[(-0.125 * N[(Pi * N[Power[f, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 6.9%

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

   2. *-commutative 6.9%

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

3. Simplified 6.9%

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

4. Taylor expanded in f around 0 96.8%

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

   1. distribute-rgt-out-- 96.8%

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

   2. metadata-eval 96.8%

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

6. Simplified 96.8%

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

7. Taylor expanded in f around 0 97.0%

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

8. Final simplification 97.0%

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

Alternative 5: 96.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* -4.0 (/ (- (log (/ 2.0 (* PI 0.5))) (log f)) PI)))

C

double code(double f) {
	return -4.0 * ((log((2.0 / (((double) M_PI) * 0.5))) - log(f)) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * ((Math.log((2.0 / (Math.PI * 0.5))) - Math.log(f)) / Math.PI);
}

Python

def code(f):
	return -4.0 * ((math.log((2.0 / (math.pi * 0.5))) - math.log(f)) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(Float64(log(Float64(2.0 / Float64(pi * 0.5))) - log(f)) / pi))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 * ((log((2.0 / (pi * 0.5))) - log(f)) / pi);
end

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 6.9%

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

   2. *-commutative 6.9%

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

3. Simplified 6.9%

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

4. Taylor expanded in f around 0 96.9%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi}} \]
5. Step-by-step derivation

   1. mul-1-neg 96.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \color{blue}{\left(-\log f\right)}}{\pi} \]

   2. unsub-neg 96.9%

      \[\leadsto -4 \cdot \frac{\color{blue}{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) - \log f}}{\pi} \]

   3. distribute-rgt-out-- 96.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{2}{\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}}\right) - \log f}{\pi} \]

   4. metadata-eval 96.9%

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

6. Simplified 96.9%

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

7. Final simplification 96.9%

   \[\leadsto -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi} \]

Alternative 6: 96.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 6.9%

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

   2. *-commutative 6.9%

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

3. Simplified 6.9%

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

4. Taylor expanded in f around 0 96.7%

   \[\leadsto \log \color{blue}{\left(\frac{2}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right)} \cdot \frac{-4}{\pi} \]
5. Step-by-step derivation

   1. associate-/r* 96.7%

      \[\leadsto \log \color{blue}{\left(\frac{\frac{2}{f}}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)} \cdot \frac{-4}{\pi} \]

   2. distribute-rgt-out-- 96.7%

      \[\leadsto \log \left(\frac{\frac{2}{f}}{\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}}\right) \cdot \frac{-4}{\pi} \]

   3. metadata-eval 96.7%

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

6. Simplified 96.7%

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

7. Taylor expanded in f around 0 96.7%

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

   1. associate-/r* 96.7%

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

9. Simplified 96.7%

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

    1. associate-*r/ 96.9%

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

    2. associate-/l/ 96.9%

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

11. Applied egg-rr 96.9%

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

    1. log1p-expm1-u 96.9%

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

    2. expm1-udef 96.9%

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

    3. add-exp-log 96.9%

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

    4. associate-/r* 96.9%

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

13. Applied egg-rr 96.9%

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

14. Final simplification 96.9%

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

Alternative 7: 95.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 6.9%

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

   2. *-commutative 6.9%

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

3. Simplified 6.9%

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

4. Taylor expanded in f around 0 96.7%

   \[\leadsto \log \color{blue}{\left(\frac{2}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right)} \cdot \frac{-4}{\pi} \]
5. Step-by-step derivation

   1. associate-/r* 96.7%

      \[\leadsto \log \color{blue}{\left(\frac{\frac{2}{f}}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)} \cdot \frac{-4}{\pi} \]

   2. distribute-rgt-out-- 96.7%

      \[\leadsto \log \left(\frac{\frac{2}{f}}{\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}}\right) \cdot \frac{-4}{\pi} \]

   3. metadata-eval 96.7%

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

6. Simplified 96.7%

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

7. Taylor expanded in f around 0 96.7%

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

   1. associate-/r* 96.7%

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

9. Simplified 96.7%

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

10. Final simplification 96.7%

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

Alternative 8: 96.0% accurate, 3.3× speedup

Math

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

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(Float64(-4.0 * 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[(N[(-4.0 * N[Log[N[(4.0 / N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

   1. distribute-lft-neg-in 6.9%

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

   2. *-commutative 6.9%

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

3. Simplified 6.9%

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

4. Taylor expanded in f around 0 96.7%

   \[\leadsto \log \color{blue}{\left(\frac{2}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right)} \cdot \frac{-4}{\pi} \]
5. Step-by-step derivation

   1. associate-/r* 96.7%

      \[\leadsto \log \color{blue}{\left(\frac{\frac{2}{f}}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)} \cdot \frac{-4}{\pi} \]

   2. distribute-rgt-out-- 96.7%

      \[\leadsto \log \left(\frac{\frac{2}{f}}{\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}}\right) \cdot \frac{-4}{\pi} \]

   3. metadata-eval 96.7%

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

6. Simplified 96.7%

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

7. Taylor expanded in f around 0 96.7%

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

   1. associate-/r* 96.7%

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

9. Simplified 96.7%

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

    1. associate-*r/ 96.9%

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

    2. associate-/l/ 96.9%

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

11. Applied egg-rr 96.9%

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

12. Final simplification 96.9%

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

Reproduce

herbie shell --seed 2023334 
(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)))))))))