VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 96.6%

Time: 26.6s

Alternatives: 6

Speedup: 3.3×

• could not determine a ground truth

  1. f = -1.6815353957630602e+158

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: 96.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \left(\left(f \cdot f\right) \cdot \left(\pi \cdot 0.041666666666666664\right)\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (fma
  (/ (- (log (/ 2.0 (* PI 0.5))) (log f)) PI)
  -4.0
  (* -2.0 (* (* f f) (* PI 0.041666666666666664)))))

C

double code(double f) {
	return fma(((log((2.0 / (((double) M_PI) * 0.5))) - log(f)) / ((double) M_PI)), -4.0, (-2.0 * ((f * f) * (((double) M_PI) * 0.041666666666666664))));
}

Julia

function code(f)
	return fma(Float64(Float64(log(Float64(2.0 / Float64(pi * 0.5))) - log(f)) / pi), -4.0, Float64(-2.0 * Float64(Float64(f * f) * Float64(pi * 0.041666666666666664))))
end

Wolfram

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

Derivation

1. Initial program 6.0%

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

   1. distribute-lft-neg-in 6.0%

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

   2. *-commutative 6.0%

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

   3. associate-/r/ 6.0%

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

   4. associate-*l/ 6.0%

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

   5. metadata-eval 6.0%

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

   6. distribute-neg-frac 6.0%

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

3. Simplified 5.9%

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

4. Taylor expanded in f around 0 95.9%

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

6. Simplified 95.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, 1 \cdot \frac{\pi}{0.5}, \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}\right), 0\right)}{\pi}, f \cdot f, 0\right)\right)} \]
7. Step-by-step derivation

   1. *-un-lft-identity 95.9%

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

   2. fma-udef 95.9%

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

   3. div-inv 95.9%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \color{blue}{\left(\pi \cdot \frac{1}{0.5}\right)} + \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

   4. metadata-eval 95.9%

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

   5. pow-div 95.9%

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

   6. metadata-eval 95.9%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \left(\pi \cdot 2\right) + \frac{-2}{{\pi}^{\color{blue}{-1}} \cdot 48}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

8. Applied egg-rr 95.9%

   \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{0.0625 \cdot \left(\pi \cdot 2\right) + \frac{-2}{{\pi}^{-1} \cdot 48}}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]
9. Step-by-step derivation

   1. *-commutative 95.9%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\left(\pi \cdot 2\right) \cdot 0.0625} + \frac{-2}{{\pi}^{-1} \cdot 48}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

   2. associate-*l* 95.9%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\pi \cdot \left(2 \cdot 0.0625\right)} + \frac{-2}{{\pi}^{-1} \cdot 48}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

   3. metadata-eval 95.9%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \pi \cdot \color{blue}{0.125} + \frac{-2}{{\pi}^{-1} \cdot 48}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

   4. associate-/r* 95.9%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \pi \cdot 0.125 + \color{blue}{\frac{\frac{-2}{{\pi}^{-1}}}{48}}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

   5. unpow-1 95.9%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \pi \cdot 0.125 + \frac{\frac{-2}{\color{blue}{\frac{1}{\pi}}}}{48}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

   6. associate-/r/ 95.9%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \pi \cdot 0.125 + \frac{\color{blue}{\frac{-2}{1} \cdot \pi}}{48}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

   7. metadata-eval 95.9%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \pi \cdot 0.125 + \frac{\color{blue}{-2} \cdot \pi}{48}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

10. Simplified 95.9%

    \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\pi \cdot 0.125 + \frac{-2 \cdot \pi}{48}}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]
11. Step-by-step derivation

    1. *-un-lft-identity 95.9%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{1 \cdot \left(\pi \cdot 0.125 + \frac{-2 \cdot \pi}{48}\right)}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

    2. fma-def 95.9%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, 1 \cdot \color{blue}{\mathsf{fma}\left(\pi, 0.125, \frac{-2 \cdot \pi}{48}\right)}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

    3. div-inv 95.9%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, 1 \cdot \mathsf{fma}\left(\pi, 0.125, \color{blue}{\left(-2 \cdot \pi\right) \cdot \frac{1}{48}}\right), 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

    4. *-commutative 95.9%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, 1 \cdot \mathsf{fma}\left(\pi, 0.125, \color{blue}{\left(\pi \cdot -2\right)} \cdot \frac{1}{48}\right), 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

    5. metadata-eval 95.9%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, 1 \cdot \mathsf{fma}\left(\pi, 0.125, \left(\pi \cdot -2\right) \cdot \color{blue}{0.020833333333333332}\right), 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

12. Applied egg-rr 95.9%

    \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{1 \cdot \mathsf{fma}\left(\pi, 0.125, \left(\pi \cdot -2\right) \cdot 0.020833333333333332\right)}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]
13. Step-by-step derivation

    1. *-lft-identity 95.9%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\mathsf{fma}\left(\pi, 0.125, \left(\pi \cdot -2\right) \cdot 0.020833333333333332\right)}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

    2. fma-udef 95.9%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\pi \cdot 0.125 + \left(\pi \cdot -2\right) \cdot 0.020833333333333332}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

    3. associate-*l* 95.9%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \pi \cdot 0.125 + \color{blue}{\pi \cdot \left(-2 \cdot 0.020833333333333332\right)}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

    4. distribute-lft-out 95.9%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\pi \cdot \left(0.125 + -2 \cdot 0.020833333333333332\right)}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

    5. metadata-eval 95.9%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \pi \cdot \left(0.125 + \color{blue}{-0.041666666666666664}\right), 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

    6. metadata-eval 95.9%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \pi \cdot \color{blue}{0.08333333333333333}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

14. Simplified 95.9%

    \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\pi \cdot 0.08333333333333333}, 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

15. Taylor expanded in f around 0 95.9%

    \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \color{blue}{\left(0.041666666666666664 \cdot \left({f}^{2} \cdot \pi\right)\right)}\right) \]
16. Step-by-step derivation

    1. *-commutative 95.9%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \color{blue}{\left(\left({f}^{2} \cdot \pi\right) \cdot 0.041666666666666664\right)}\right) \]

    2. associate-*l* 95.9%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \color{blue}{\left({f}^{2} \cdot \left(\pi \cdot 0.041666666666666664\right)\right)}\right) \]

    3. unpow2 95.9%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \left(\color{blue}{\left(f \cdot f\right)} \cdot \left(\pi \cdot 0.041666666666666664\right)\right)\right) \]

17. Simplified 95.9%

    \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \color{blue}{\left(\left(f \cdot f\right) \cdot \left(\pi \cdot 0.041666666666666664\right)\right)}\right) \]

18. Final simplification 95.9%

    \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -4, -2 \cdot \left(\left(f \cdot f\right) \cdot \left(\pi \cdot 0.041666666666666664\right)\right)\right) \]

Alternative 2: 96.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -\mathsf{fma}\left(\pi \cdot \left(f \cdot f\right), 0.125, \frac{4 \cdot \log \left(\frac{4}{\pi \cdot f}\right)}{\pi}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (- (fma (* PI (* f f)) 0.125 (/ (* 4.0 (log (/ 4.0 (* PI f)))) PI))))

C

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

Julia

function code(f)
	return Float64(-fma(Float64(pi * Float64(f * f)), 0.125, Float64(Float64(4.0 * log(Float64(4.0 / Float64(pi * f)))) / pi)))
end

Wolfram

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

Derivation

1. Initial program 6.0%

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

2. Taylor expanded in f around 0 95.4%

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

   1. distribute-rgt-out-- 95.4%

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

   2. metadata-eval 95.4%

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

4. Simplified 95.4%

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

5. Taylor expanded in f around 0 95.7%

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

   1. *-commutative 95.7%

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

   2. fma-def 95.7%

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

   3. unpow2 95.7%

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

   4. associate-*r/ 95.7%

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

7. Simplified 95.6%

   \[\leadsto -\color{blue}{\mathsf{fma}\left(\left(f \cdot f\right) \cdot \pi, 0.125, \frac{4 \cdot \log \left(\frac{4}{f \cdot \pi}\right)}{\pi}\right)} \]

8. Final simplification 95.6%

   \[\leadsto -\mathsf{fma}\left(\pi \cdot \left(f \cdot f\right), 0.125, \frac{4 \cdot \log \left(\frac{4}{\pi \cdot f}\right)}{\pi}\right) \]

Alternative 3: 95.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.0%

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

   1. distribute-lft-neg-in 6.0%

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

   2. *-commutative 6.0%

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

   3. associate-/r/ 6.0%

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

   4. associate-*l/ 6.0%

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

   5. metadata-eval 6.0%

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

   6. distribute-neg-frac 6.0%

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

3. Simplified 5.9%

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

4. Taylor expanded in f around 0 95.5%

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

   1. *-commutative 95.5%

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

   2. mul-1-neg 95.5%

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

   3. unsub-neg 95.5%

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

   4. distribute-rgt-out-- 95.5%

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

   5. metadata-eval 95.5%

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

   6. associate-/r* 95.5%

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

6. Simplified 95.5%

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

7. Taylor expanded in f around 0 95.5%

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

   1. log-div 95.5%

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

   2. associate--l- 95.4%

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

   3. log-prod 95.4%

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

   4. *-commutative 95.4%

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

   5. log-div 95.5%

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

   6. metadata-eval 95.5%

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

   7. associate-/r* 95.5%

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

   8. associate-*r* 95.5%

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

   9. *-commutative 95.5%

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

   10. *-commutative 95.5%

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

   11. associate-*r* 95.5%

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

   12. associate-/r* 95.5%

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

   13. metadata-eval 95.5%

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

   14. associate-/r* 95.5%

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

9. Simplified 95.5%

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

    1. log-div 95.5%

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

11. Applied egg-rr 95.5%

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

12. Final simplification 95.5%

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

Alternative 4: 96.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.0%

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

   1. distribute-lft-neg-in 6.0%

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

   2. *-commutative 6.0%

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

   3. associate-/r/ 6.0%

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

   4. associate-*l/ 6.0%

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

   5. metadata-eval 6.0%

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

   6. distribute-neg-frac 6.0%

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

3. Simplified 5.9%

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

4. Taylor expanded in f around 0 95.5%

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

   1. *-commutative 95.5%

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

   2. mul-1-neg 95.5%

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

   3. unsub-neg 95.5%

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

   4. distribute-rgt-out-- 95.5%

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

   5. metadata-eval 95.5%

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

   6. associate-/r* 95.5%

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

6. Simplified 95.5%

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

7. Taylor expanded in f around 0 95.5%

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

8. Final simplification 95.5%

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

Alternative 5: 95.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.0%

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

   1. distribute-lft-neg-in 6.0%

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

   2. *-commutative 6.0%

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

   3. associate-/r/ 6.0%

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

   4. associate-*l/ 6.0%

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

   5. metadata-eval 6.0%

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

   6. distribute-neg-frac 6.0%

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

3. Simplified 5.9%

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

4. Taylor expanded in f around 0 95.5%

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

   1. *-commutative 95.5%

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

   2. mul-1-neg 95.5%

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

   3. unsub-neg 95.5%

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

   4. distribute-rgt-out-- 95.5%

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

   5. metadata-eval 95.5%

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

   6. associate-/r* 95.5%

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

6. Simplified 95.5%

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

7. Taylor expanded in f around 0 95.5%

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

   1. log-div 95.5%

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

   2. associate--l- 95.4%

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

   3. log-prod 95.4%

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

   4. *-commutative 95.4%

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

   5. log-div 95.5%

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

   6. metadata-eval 95.5%

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

   7. associate-/r* 95.5%

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

   8. associate-*r* 95.5%

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

   9. *-commutative 95.5%

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

   10. *-commutative 95.5%

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

   11. associate-*r* 95.5%

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

   12. associate-/r* 95.5%

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

   13. metadata-eval 95.5%

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

   14. associate-/r* 95.5%

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

9. Simplified 95.5%

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

10. Final simplification 95.5%

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

Alternative 6: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log 0.3333333333333333}{\pi} \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ (log 0.3333333333333333) PI) (- 4.0)))

C

double code(double f) {
	return (log(0.3333333333333333) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return (Math.log(0.3333333333333333) / Math.PI) * -4.0;
}

Python

def code(f):
	return (math.log(0.3333333333333333) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(log(0.3333333333333333) / pi) * Float64(-4.0))
end

MATLAB

function tmp = code(f)
	tmp = (log(0.3333333333333333) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[Log[0.3333333333333333], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision]

Derivation

1. Initial program 6.0%

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

3. Applied egg-rr 1.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{6}}\right) \]

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.3333333333333333}{\pi}} \]

5. Final simplification 1.6%

   \[\leadsto \frac{\log 0.3333333333333333}{\pi} \cdot \left(-4\right) \]

Reproduce

herbie shell --seed 2023200 
(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)))))))))