VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 99.0%

Time: 31.4s

Alternatives: 6

Speedup: 4.7×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{2 \cdot \log \left(\sqrt{\tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right)}\right)}{-\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* -4.0 (/ (* 2.0 (log (sqrt (tanh (* 0.25 (* f PI)))))) (- PI))))

C

double code(double f) {
	return -4.0 * ((2.0 * log(sqrt(tanh((0.25 * (f * ((double) M_PI))))))) / -((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * ((2.0 * Math.log(Math.sqrt(Math.tanh((0.25 * (f * Math.PI)))))) / -Math.PI);
}

Python

def code(f):
	return -4.0 * ((2.0 * math.log(math.sqrt(math.tanh((0.25 * (f * math.pi)))))) / -math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(Float64(2.0 * log(sqrt(tanh(Float64(0.25 * Float64(f * pi)))))) / Float64(-pi)))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 * ((2.0 * log(sqrt(tanh((0.25 * (f * pi)))))) / -pi);
end

Wolfram

code[f_] := N[(-4.0 * N[(N[(2.0 * N[Log[N[Sqrt[N[Tanh[N[(0.25 * N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-Pi)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

   1. associate-*l/ 6.5%

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

4. Applied egg-rr 99.4%

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

   1. neg-mul-1 99.4%

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

   2. *-commutative 99.4%

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

   3. times-frac 99.4%

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

   4. metadata-eval 99.4%

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

   5. *-commutative 99.4%

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

   6. *-commutative 99.4%

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

6. Simplified 99.4%

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

8. Applied egg-rr 99.4%

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

   1. count-2 99.4%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

    \[\leadsto -4 \cdot \frac{2 \cdot \log \left(\sqrt{\tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right)}\right)}{-\pi} \]
12. Add Preprocessing

Alternative 2: 97.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if f < 1.25

   1. Initial program 6.2%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in f around 0 98.4%

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

      1. associate-*r/ 98.4%

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

      2. mul-1-neg 98.4%

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

      3. unsub-neg 98.4%

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

   7. Simplified 98.4%

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

   if 1.25 < f

   1. Initial program 16.7%

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

      1. associate-*l/ 16.7%

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

   4. Applied egg-rr 95.0%

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

      1. neg-mul-1 95.0%

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

      2. *-commutative 95.0%

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

      3. times-frac 95.0%

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

      4. metadata-eval 95.0%

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

      5. *-commutative 95.0%

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

      6. *-commutative 95.0%

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

   6. Simplified 95.0%

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

   8. Applied egg-rr 78.5%

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

      1. *-inverses 78.5%

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

   10. Simplified 78.5%

       \[\leadsto --4 \cdot \frac{\log \color{blue}{1}}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

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

Alternative 3: 99.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \tanh \left(f \cdot \left(0.25 \cdot \pi\right)\right)}{-\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* -4.0 (/ (log (tanh (* f (* 0.25 PI)))) (- PI))))

C

double code(double f) {
	return -4.0 * (log(tanh((f * (0.25 * ((double) M_PI))))) / -((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log(Math.tanh((f * (0.25 * Math.PI)))) / -Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log(math.tanh((f * (0.25 * math.pi)))) / -math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log(tanh(Float64(f * Float64(0.25 * pi)))) / Float64(-pi)))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 * (log(tanh((f * (0.25 * pi)))) / -pi);
end

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. associate-*l/ 6.5%

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

4. Applied egg-rr 99.4%

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

   1. neg-mul-1 99.4%

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

   2. *-commutative 99.4%

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

   3. times-frac 99.4%

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

   4. metadata-eval 99.4%

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

   5. *-commutative 99.4%

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

   6. *-commutative 99.4%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

   \[\leadsto -4 \cdot \frac{\log \tanh \left(f \cdot \left(0.25 \cdot \pi\right)\right)}{-\pi} \]
8. Add Preprocessing

Alternative 4: 97.6% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if f < 1.25

   1. Initial program 6.2%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in f around 0 98.4%

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

      1. associate-*r/ 98.4%

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

      2. mul-1-neg 98.4%

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

      3. unsub-neg 98.4%

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

   7. Simplified 98.4%

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

      1. associate-/l* 98.4%

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

      2. diff-log 98.3%

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

   9. Applied egg-rr 98.3%

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

       1. associate-*r/ 98.3%

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

       2. metadata-eval 98.3%

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

       3. distribute-lft-neg-in 98.3%

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

       4. *-commutative 98.3%

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

       5. distribute-rgt-neg-in 98.3%

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

       6. metadata-eval 98.3%

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

       7. associate-*r/ 98.2%

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

       8. associate-/l/ 98.2%

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

   11. Simplified 98.2%

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

   if 1.25 < f

   1. Initial program 16.7%

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

      1. associate-*l/ 16.7%

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

   4. Applied egg-rr 95.0%

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

      1. neg-mul-1 95.0%

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

      2. *-commutative 95.0%

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

      3. times-frac 95.0%

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

      4. metadata-eval 95.0%

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

      5. *-commutative 95.0%

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

      6. *-commutative 95.0%

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

   6. Simplified 95.0%

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

   8. Applied egg-rr 78.5%

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

      1. *-inverses 78.5%

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

   10. Simplified 78.5%

       \[\leadsto --4 \cdot \frac{\log \color{blue}{1}}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

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

Alternative 5: 97.8% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if f < 1.25

   1. Initial program 6.2%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in f around 0 98.4%

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

      1. associate-*r/ 98.4%

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

      2. mul-1-neg 98.4%

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

      3. unsub-neg 98.4%

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

   7. Simplified 98.4%

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

      1. diff-log 98.3%

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

   9. Applied egg-rr 98.3%

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

   if 1.25 < f

   1. Initial program 16.7%

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

      1. associate-*l/ 16.7%

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

   4. Applied egg-rr 95.0%

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

      1. neg-mul-1 95.0%

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

      2. *-commutative 95.0%

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

      3. times-frac 95.0%

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

      4. metadata-eval 95.0%

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

      5. *-commutative 95.0%

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

      6. *-commutative 95.0%

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

   6. Simplified 95.0%

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

   8. Applied egg-rr 78.5%

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

      1. *-inverses 78.5%

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

   10. Simplified 78.5%

       \[\leadsto --4 \cdot \frac{\log \color{blue}{1}}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

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

Alternative 6: 5.1% accurate, 106.4× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{0}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* -4.0 (/ 0.0 PI)))

C

double code(double f) {
	return -4.0 * (0.0 / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (0.0 / Math.PI);
}

Python

def code(f):
	return -4.0 * (0.0 / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(0.0 / pi))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 * (0.0 / pi);
end

Wolfram

code[f_] := N[(-4.0 * N[(0.0 / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

   1. associate-*l/ 6.5%

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

4. Applied egg-rr 99.4%

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

   1. neg-mul-1 99.4%

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

   2. *-commutative 99.4%

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

   3. times-frac 99.4%

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

   4. metadata-eval 99.4%

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

   5. *-commutative 99.4%

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

   6. *-commutative 99.4%

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

6. Simplified 99.4%

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

8. Applied egg-rr 5.8%

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

   1. *-inverses 5.8%

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

10. Simplified 5.8%

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

11. Final simplification 5.8%

    \[\leadsto -4 \cdot \frac{0}{\pi} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(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)))))))))