VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.2% → 96.0%

Time: 31.8s

Alternatives: 5

Speedup: 4.8×

• 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: 7.2% 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: 96.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.8%

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

3. Taylor expanded in f around 0 96.7%

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

   1. associate-*r/ 96.7%

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

   2. mul-1-neg 96.7%

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

   3. unsub-neg 96.7%

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

   4. distribute-rgt-out-- 96.7%

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

   5. metadata-eval 96.7%

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

5. Simplified 96.7%

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

6. Final simplification 96.7%

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

Alternative 2: 96.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(4 \cdot \left(0.03125 \cdot \left(\pi \cdot f\right) + \frac{1}{\pi \cdot f}\right)\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (- (log (* 4.0 (+ (* 0.03125 (* PI f)) (/ 1.0 (* PI f)))))) (* PI 0.25)))

C

double code(double f) {
	return -log((4.0 * ((0.03125 * (((double) M_PI) * f)) + (1.0 / (((double) M_PI) * f))))) / (((double) M_PI) * 0.25);
}

Java

public static double code(double f) {
	return -Math.log((4.0 * ((0.03125 * (Math.PI * f)) + (1.0 / (Math.PI * f))))) / (Math.PI * 0.25);
}

Python

def code(f):
	return -math.log((4.0 * ((0.03125 * (math.pi * f)) + (1.0 / (math.pi * f))))) / (math.pi * 0.25)

Julia

function code(f)
	return Float64(Float64(-log(Float64(4.0 * Float64(Float64(0.03125 * Float64(pi * f)) + Float64(1.0 / Float64(pi * f)))))) / Float64(pi * 0.25))
end

MATLAB

function tmp = code(f)
	tmp = -log((4.0 * ((0.03125 * (pi * f)) + (1.0 / (pi * f))))) / (pi * 0.25);
end

Wolfram

code[f_] := N[((-N[Log[N[(4.0 * N[(N[(0.03125 * N[(Pi * f), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.8%

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

3. Taylor expanded in f around 0 96.4%

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

   1. distribute-rgt-out-- 96.4%

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

   2. metadata-eval 96.4%

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

5. Simplified 96.4%

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

   1. associate-*l/ 96.6%

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

   2. *-un-lft-identity 96.6%

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

   3. cosh-undef 96.6%

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

   4. associate-*l/ 96.6%

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

   5. div-inv 96.6%

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

   6. metadata-eval 96.6%

      \[\leadsto -\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{f \cdot \left(\pi \cdot 0.5\right)}\right)}{\pi \cdot \color{blue}{0.25}} \]

7. Applied egg-rr 96.6%

   \[\leadsto -\color{blue}{\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{f \cdot \left(\pi \cdot 0.5\right)}\right)}{\pi \cdot 0.25}} \]
8. Step-by-step derivation

   1. associate-*r* 96.6%

      \[\leadsto -\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{\color{blue}{\left(f \cdot \pi\right) \cdot 0.5}}\right)}{\pi \cdot 0.25} \]

   2. *-commutative 96.6%

      \[\leadsto -\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{\color{blue}{0.5 \cdot \left(f \cdot \pi\right)}}\right)}{\pi \cdot 0.25} \]

   3. times-frac 96.6%

      \[\leadsto -\frac{\log \color{blue}{\left(\frac{2}{0.5} \cdot \frac{\cosh \left(\frac{\pi \cdot f}{4}\right)}{f \cdot \pi}\right)}}{\pi \cdot 0.25} \]

   4. metadata-eval 96.6%

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

   5. associate-/l* 96.6%

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

   6. *-commutative 96.6%

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

9. Simplified 96.6%

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

10. Taylor expanded in f around 0 96.6%

    \[\leadsto -\frac{\log \left(4 \cdot \color{blue}{\left(0.03125 \cdot \left(f \cdot \pi\right) + \frac{1}{f \cdot \pi}\right)}\right)}{\pi \cdot 0.25} \]

11. Final simplification 96.6%

    \[\leadsto \frac{-\log \left(4 \cdot \left(0.03125 \cdot \left(\pi \cdot f\right) + \frac{1}{\pi \cdot f}\right)\right)}{\pi \cdot 0.25} \]
12. Add Preprocessing

Alternative 3: 95.8% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.8%

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

3. Taylor expanded in f around 0 96.7%

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

   1. associate-*r/ 96.7%

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

   2. associate-/l* 96.5%

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

   3. associate-/r/ 96.6%

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

   4. mul-1-neg 96.6%

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

   5. unsub-neg 96.6%

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

   6. distribute-rgt-out-- 96.6%

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

   7. *-commutative 96.6%

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

   8. associate-/r* 96.6%

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

   9. metadata-eval 96.6%

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

   10. metadata-eval 96.6%

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

5. Simplified 96.6%

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

   1. diff-log 96.4%

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

   2. associate-/l/ 96.4%

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

7. Applied egg-rr 96.4%

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

8. Final simplification 96.4%

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

Alternative 4: 95.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.8%

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

3. Taylor expanded in f around 0 96.7%

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

   1. associate-*r/ 96.7%

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

   2. mul-1-neg 96.7%

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

   3. unsub-neg 96.7%

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

   4. distribute-rgt-out-- 96.7%

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

   5. metadata-eval 96.7%

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

5. Simplified 96.7%

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

6. Taylor expanded in f around 0 96.7%

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

   1. log-div 96.7%

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

   2. associate--l- 96.5%

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

   3. log-prod 96.5%

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

   4. *-commutative 96.5%

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

   5. log-div 96.6%

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

   6. *-commutative 96.6%

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

8. Simplified 96.6%

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

9. Final simplification 96.6%

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

Alternative 5: 0.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(-\log 0\right)}{\pi} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.8%

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

3. Taylor expanded in f around 0 96.4%

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

4. Taylor expanded in f around inf 0.7%

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

   1. associate-*r/ 0.7%

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

   2. distribute-rgt-out 0.7%

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

   3. distribute-rgt-out-- 0.7%

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

   4. metadata-eval 0.7%

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

   5. metadata-eval 0.7%

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

   6. mul0-rgt 0.7%

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

6. Simplified 0.7%

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

7. Final simplification 0.7%

   \[\leadsto \frac{4 \cdot \left(-\log 0\right)}{\pi} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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)))))))))