VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 98.9%

Time: 29.4s

Alternatives: 5

Speedup: 532.0×

• 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: 98.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}{0.25}}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (/ (log (tanh (* (* 0.25 f) PI))) 0.25) PI))

C

double code(double f) {
	return (log(tanh(((0.25 * f) * ((double) M_PI)))) / 0.25) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (Math.log(Math.tanh(((0.25 * f) * Math.PI))) / 0.25) / Math.PI;
}

Python

def code(f):
	return (math.log(math.tanh(((0.25 * f) * math.pi))) / 0.25) / math.pi

Julia

function code(f)
	return Float64(Float64(log(tanh(Float64(Float64(0.25 * f) * pi))) / 0.25) / pi)
end

MATLAB

function tmp = code(f)
	tmp = (log(tanh(((0.25 * f) * pi))) / 0.25) / pi;
end

Wolfram

code[f_] := N[(N[(N[Log[N[Tanh[N[(N[(0.25 * f), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / 0.25), $MachinePrecision] / Pi), $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. distribute-neg-frac2 6.0%

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

   3. associate-*l/ 6.0%

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

3. Simplified 6.0%

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

   1. add-cube-cbrt 6.0%

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

   2. log-prod 6.0%

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

6. Applied egg-rr 6.0%

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

8. Simplified 99.1%

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

   1. *-un-lft-identity 99.1%

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

   2. div-inv 99.1%

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

   3. times-frac 99.0%

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

10. Applied egg-rr 99.2%

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

    1. associate-*l/ 99.3%

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

    2. *-lft-identity 99.3%

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

    3. distribute-frac-neg 99.3%

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

    4. distribute-neg-frac2 99.3%

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

    5. associate-*r* 99.3%

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

    6. *-commutative 99.3%

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

    7. associate-*r* 99.3%

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

    8. metadata-eval 99.3%

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

12. Simplified 99.3%

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

13. Final simplification 99.3%

    \[\leadsto \frac{\frac{\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}{0.25}}{\pi} \]
14. Add Preprocessing

Alternative 2: 97.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.3:\\ \;\;\;\;\log \left(0.25 \cdot \left(f \cdot \pi\right)\right) \cdot \frac{-4}{-\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.3) (* (log (* 0.25 (* f PI))) (/ -4.0 (- PI))) 0.0))

C

double code(double f) {
	double tmp;
	if (f <= 1.3) {
		tmp = log((0.25 * (f * ((double) M_PI)))) * (-4.0 / -((double) M_PI));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.3) {
		tmp = Math.log((0.25 * (f * Math.PI))) * (-4.0 / -Math.PI);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.3:
		tmp = math.log((0.25 * (f * math.pi))) * (-4.0 / -math.pi)
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.3)
		tmp = Float64(log(Float64(0.25 * Float64(f * pi))) * Float64(-4.0 / Float64(-pi)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.3)
		tmp = log((0.25 * (f * pi))) * (-4.0 / -pi);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.3], N[(N[Log[N[(0.25 * N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / (-Pi)), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 1.30000000000000004

   1. Initial program 6.1%

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

   3. Taylor expanded in f around 0 98.5%

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

      1. distribute-rgt-out-- 98.5%

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

      2. metadata-eval 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in f around 0 98.5%

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

      1. div-inv 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*l/ 98.7%

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

      4. *-un-lft-identity 98.7%

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

      5. metadata-eval 98.7%

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

      6. div-inv 98.7%

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

      7. frac-2neg 98.7%

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

      8. metadata-eval 98.7%

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

      9. associate-/r/ 98.7%

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

      10. clear-num 98.7%

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

      11. log-rec 98.7%

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

      12. div-inv 98.7%

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

      13. metadata-eval 98.7%

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

      14. associate-*r* 98.7%

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

      15. frac-2neg 98.7%

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

   8. Applied egg-rr 98.5%

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

      1. associate-/r/ 98.5%

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

      2. associate-*r* 98.5%

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

      3. *-commutative 98.5%

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

      4. associate-*r* 98.5%

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

      5. associate-*l/ 98.5%

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

      6. metadata-eval 98.5%

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

   10. Simplified 98.5%

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

   if 1.30000000000000004 < f

   1. Initial program 5.2%

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

      1. distribute-lft-neg-in 5.2%

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

      2. distribute-neg-frac2 5.2%

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

      3. associate-*l/ 5.2%

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

   3. Simplified 5.2%

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

   5. Taylor expanded in f around inf 5.2%

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

   7. Simplified 93.3%

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

   9. Applied egg-rr 87.9%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.3:\\ \;\;\;\;\log \left(0.25 \cdot \left(f \cdot \pi\right)\right) \cdot \frac{-4}{-\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.3:\\ \;\;\;\;\frac{-4 \cdot \log \left(f \cdot \left(0.25 \cdot \pi\right)\right)}{-\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.3) (/ (* -4.0 (log (* f (* 0.25 PI)))) (- PI)) 0.0))

C

double code(double f) {
	double tmp;
	if (f <= 1.3) {
		tmp = (-4.0 * log((f * (0.25 * ((double) M_PI))))) / -((double) M_PI);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.3) {
		tmp = (-4.0 * Math.log((f * (0.25 * Math.PI)))) / -Math.PI;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.3:
		tmp = (-4.0 * math.log((f * (0.25 * math.pi)))) / -math.pi
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.3)
		tmp = Float64(Float64(-4.0 * log(Float64(f * Float64(0.25 * pi)))) / Float64(-pi));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.3)
		tmp = (-4.0 * log((f * (0.25 * pi)))) / -pi;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.3], N[(N[(-4.0 * N[Log[N[(f * N[(0.25 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-Pi)), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 1.30000000000000004

   1. Initial program 6.1%

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

      1. add-sqr-sqrt 6.1%

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

      2. pow2 6.1%

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in f around 0 98.3%

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

      1. unpow2 98.3%

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

      2. add-sqr-sqrt 98.7%

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

      3. associate-*r/ 98.7%

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

      4. sum-log 98.7%

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

      5. *-commutative 98.7%

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

   7. Applied egg-rr 98.7%

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

   if 1.30000000000000004 < f

   1. Initial program 5.2%

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

      1. distribute-lft-neg-in 5.2%

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

      2. distribute-neg-frac2 5.2%

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

      3. associate-*l/ 5.2%

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

   3. Simplified 5.2%

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

   5. Taylor expanded in f around inf 5.2%

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

   7. Simplified 93.3%

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

   9. Applied egg-rr 87.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.3:\\ \;\;\;\;\frac{-4 \cdot \log \left(f \cdot \left(0.25 \cdot \pi\right)\right)}{-\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 17.6% accurate, 88.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 230:\\ \;\;\;\;-16\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (if (<= f 230.0) -16.0 0.0))

C

double code(double f) {
	double tmp;
	if (f <= 230.0) {
		tmp = -16.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 230.0d0) then
        tmp = -16.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double f) {
	double tmp;
	if (f <= 230.0) {
		tmp = -16.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 230.0:
		tmp = -16.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 230.0)
		tmp = -16.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 230.0)
		tmp = -16.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 230.0], -16.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 230

   1. Initial program 6.2%

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

      1. distribute-lft-neg-in 6.2%

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

      2. distribute-neg-frac2 6.2%

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

      3. associate-*l/ 6.2%

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

   3. Simplified 6.2%

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

   5. Taylor expanded in f around inf 6.2%

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

   7. Simplified 99.3%

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

   9. Applied egg-rr 15.6%

      \[\leadsto \color{blue}{-16} \]

   if 230 < f

   1. Initial program 0.0%

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

      1. distribute-lft-neg-in 0.0%

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

      2. distribute-neg-frac2 0.0%

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

      3. associate-*l/ 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in f around inf 0.0%

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

   7. Simplified 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 17.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 230:\\ \;\;\;\;-16\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 15.3% accurate, 532.0× speedup

Math

\[\begin{array}{l} \\ -16 \end{array} \]

FPCore

(FPCore (f) :precision binary64 -16.0)

C

double code(double f) {
	return -16.0;
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = -16.0d0
end function

Java

public static double code(double f) {
	return -16.0;
}

Python

def code(f):
	return -16.0

Julia

function code(f)
	return -16.0
end

MATLAB

function tmp = code(f)
	tmp = -16.0;
end

Wolfram

code[f_] := -16.0

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. distribute-neg-frac2 6.0%

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

   3. associate-*l/ 6.0%

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

3. Simplified 6.0%

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

5. Taylor expanded in f around inf 6.0%

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

7. Simplified 99.3%

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

9. Applied egg-rr 15.2%

   \[\leadsto \color{blue}{-16} \]

10. Final simplification 15.2%

    \[\leadsto -16 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(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)))))))))