VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 99.1%

Time: 29.6s

Alternatives: 6

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: 6.9% 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: 99.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

3. Taylor expanded in f around inf 6.4%

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

   1. clear-num 6.4%

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

   2. log-rec 6.4%

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

   3. +-commutative 6.4%

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

   4. tanh-undef 98.4%

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

   5. associate-*r* 98.4%

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

5. Applied egg-rr 98.4%

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

6. Final simplification 98.4%

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

Alternative 2: 97.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.5:\\ \;\;\;\;4 \cdot \frac{-\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right)}{\pi} \cdot \left(-4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.5)
   (* 4.0 (/ (- (log (/ (/ 4.0 PI) f))) PI))
   (* (/ (log (tanh (* 0.25 (* f PI)))) PI) (- 4.0))))

C

double code(double f) {
	double tmp;
	if (f <= 1.5) {
		tmp = 4.0 * (-log(((4.0 / ((double) M_PI)) / f)) / ((double) M_PI));
	} else {
		tmp = (log(tanh((0.25 * (f * ((double) M_PI))))) / ((double) M_PI)) * -4.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.5) {
		tmp = 4.0 * (-Math.log(((4.0 / Math.PI) / f)) / Math.PI);
	} else {
		tmp = (Math.log(Math.tanh((0.25 * (f * Math.PI)))) / Math.PI) * -4.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.5:
		tmp = 4.0 * (-math.log(((4.0 / math.pi) / f)) / math.pi)
	else:
		tmp = (math.log(math.tanh((0.25 * (f * math.pi)))) / math.pi) * -4.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.5)
		tmp = Float64(4.0 * Float64(Float64(-log(Float64(Float64(4.0 / pi) / f))) / pi));
	else
		tmp = Float64(Float64(log(tanh(Float64(0.25 * Float64(f * pi)))) / pi) * Float64(-4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.5)
		tmp = 4.0 * (-log(((4.0 / pi) / f)) / pi);
	else
		tmp = (log(tanh((0.25 * (f * pi)))) / pi) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.5], N[(4.0 * N[((-N[Log[N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision]) / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[Tanh[N[(0.25 * N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < 1.5

   1. Initial program 6.6%

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

   3. Taylor expanded in f around 0 98.2%

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

      1. *-commutative 98.2%

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

      2. associate-/r* 98.2%

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

      3. distribute-rgt-out-- 98.2%

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

      4. metadata-eval 98.2%

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

      5. associate-/r* 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in f around 0 98.4%

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

      1. neg-mul-1 98.4%

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

      2. sub-neg 98.4%

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

   8. Simplified 98.4%

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

   9. Taylor expanded in f around inf 98.4%

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

       1. log-rec 98.4%

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

       2. mul-1-neg 98.4%

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

       3. remove-double-neg 98.4%

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

       4. log-div 98.3%

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

   11. Simplified 98.3%

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

   if 1.5 < f

   1. Initial program 1.5%

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

   3. Taylor expanded in f around inf 1.5%

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

      1. clear-num 1.5%

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

      2. log-rec 1.5%

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

      3. +-commutative 1.5%

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

      4. tanh-undef 71.5%

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

      5. associate-*r* 71.5%

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

   5. Applied egg-rr 71.5%

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

      1. expm1-log1p-u 71.5%

         \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}{\pi}\right)\right)} \]

      2. expm1-udef 71.5%

         \[\leadsto -4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}{\pi}\right)} - 1\right)} \]

      3. add-sqr-sqrt 71.5%

         \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)} \cdot \sqrt{-\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}}}{\pi}\right)} - 1\right) \]

      4. sqrt-unprod 71.5%

         \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)\right) \cdot \left(-\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)\right)}}}{\pi}\right)} - 1\right) \]

      5. sqr-neg 71.5%

         \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right) \cdot \log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}}}{\pi}\right)} - 1\right) \]

      6. sqrt-unprod 71.5%

         \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)} \cdot \sqrt{\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}}}{\pi}\right)} - 1\right) \]

      7. add-sqr-sqrt 71.5%

         \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}}{\pi}\right)} - 1\right) \]

      8. associate-*l* 71.5%

         \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\log \tanh \color{blue}{\left(0.25 \cdot \left(f \cdot \pi\right)\right)}}{\pi}\right)} - 1\right) \]

   7. Applied egg-rr 71.5%

      \[\leadsto -4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right)}{\pi}\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 71.5%

         \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right)}{\pi}\right)\right)} \]

      2. expm1-log1p 71.5%

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

   9. Simplified 71.5%

      \[\leadsto -4 \cdot \color{blue}{\frac{\log \tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right)}{\pi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.5:\\ \;\;\;\;4 \cdot \frac{-\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right)}{\pi} \cdot \left(-4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.5:\\ \;\;\;\;4 \cdot \frac{\log f - \log \left(\frac{4}{\pi}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right)}{\pi} \cdot \left(-4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.5)
   (* 4.0 (/ (- (log f) (log (/ 4.0 PI))) PI))
   (* (/ (log (tanh (* 0.25 (* f PI)))) PI) (- 4.0))))

C

double code(double f) {
	double tmp;
	if (f <= 1.5) {
		tmp = 4.0 * ((log(f) - log((4.0 / ((double) M_PI)))) / ((double) M_PI));
	} else {
		tmp = (log(tanh((0.25 * (f * ((double) M_PI))))) / ((double) M_PI)) * -4.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.5) {
		tmp = 4.0 * ((Math.log(f) - Math.log((4.0 / Math.PI))) / Math.PI);
	} else {
		tmp = (Math.log(Math.tanh((0.25 * (f * Math.PI)))) / Math.PI) * -4.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.5:
		tmp = 4.0 * ((math.log(f) - math.log((4.0 / math.pi))) / math.pi)
	else:
		tmp = (math.log(math.tanh((0.25 * (f * math.pi)))) / math.pi) * -4.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.5)
		tmp = Float64(4.0 * Float64(Float64(log(f) - log(Float64(4.0 / pi))) / pi));
	else
		tmp = Float64(Float64(log(tanh(Float64(0.25 * Float64(f * pi)))) / pi) * Float64(-4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.5)
		tmp = 4.0 * ((log(f) - log((4.0 / pi))) / pi);
	else
		tmp = (log(tanh((0.25 * (f * pi)))) / pi) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.5], N[(4.0 * N[(N[(N[Log[f], $MachinePrecision] - N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[Tanh[N[(0.25 * N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < 1.5

   1. Initial program 6.6%

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

   3. Taylor expanded in f around 0 98.2%

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

      1. *-commutative 98.2%

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

      2. associate-/r* 98.2%

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

      3. distribute-rgt-out-- 98.2%

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

      4. metadata-eval 98.2%

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

      5. associate-/r* 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in f around 0 98.4%

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

      1. neg-mul-1 98.4%

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

      2. sub-neg 98.4%

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

   8. Simplified 98.4%

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

   if 1.5 < f

   1. Initial program 1.5%

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

   3. Taylor expanded in f around inf 1.5%

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

      1. clear-num 1.5%

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

      2. log-rec 1.5%

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

      3. +-commutative 1.5%

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

      4. tanh-undef 71.5%

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

      5. associate-*r* 71.5%

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

   5. Applied egg-rr 71.5%

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

      1. expm1-log1p-u 71.5%

         \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}{\pi}\right)\right)} \]

      2. expm1-udef 71.5%

         \[\leadsto -4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}{\pi}\right)} - 1\right)} \]

      3. add-sqr-sqrt 71.5%

         \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)} \cdot \sqrt{-\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}}}{\pi}\right)} - 1\right) \]

      4. sqrt-unprod 71.5%

         \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)\right) \cdot \left(-\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)\right)}}}{\pi}\right)} - 1\right) \]

      5. sqr-neg 71.5%

         \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right) \cdot \log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}}}{\pi}\right)} - 1\right) \]

      6. sqrt-unprod 71.5%

         \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)} \cdot \sqrt{\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}}}{\pi}\right)} - 1\right) \]

      7. add-sqr-sqrt 71.5%

         \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}}{\pi}\right)} - 1\right) \]

      8. associate-*l* 71.5%

         \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\log \tanh \color{blue}{\left(0.25 \cdot \left(f \cdot \pi\right)\right)}}{\pi}\right)} - 1\right) \]

   7. Applied egg-rr 71.5%

      \[\leadsto -4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right)}{\pi}\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 71.5%

         \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right)}{\pi}\right)\right)} \]

      2. expm1-log1p 71.5%

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

   9. Simplified 71.5%

      \[\leadsto -4 \cdot \color{blue}{\frac{\log \tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right)}{\pi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.5:\\ \;\;\;\;4 \cdot \frac{\log f - \log \left(\frac{4}{\pi}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right)}{\pi} \cdot \left(-4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 1.6% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

3. Taylor expanded in f around inf 6.4%

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

   1. clear-num 6.4%

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

   2. log-rec 6.4%

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

   3. +-commutative 6.4%

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

   4. tanh-undef 98.4%

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

   5. associate-*r* 98.4%

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

5. Applied egg-rr 98.4%

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

   1. expm1-log1p-u 97.2%

      \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}{\pi}\right)\right)} \]

   2. expm1-udef 97.2%

      \[\leadsto -4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}{\pi}\right)} - 1\right)} \]

   3. add-sqr-sqrt 97.1%

      \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)} \cdot \sqrt{-\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}}}{\pi}\right)} - 1\right) \]

   4. sqrt-unprod 97.2%

      \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)\right) \cdot \left(-\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)\right)}}}{\pi}\right)} - 1\right) \]

   5. sqr-neg 97.2%

      \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right) \cdot \log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}}}{\pi}\right)} - 1\right) \]

   6. sqrt-unprod 2.8%

      \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)} \cdot \sqrt{\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}}}{\pi}\right)} - 1\right) \]

   7. add-sqr-sqrt 2.8%

      \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\log \tanh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}}{\pi}\right)} - 1\right) \]

   8. associate-*l* 2.8%

      \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\log \tanh \color{blue}{\left(0.25 \cdot \left(f \cdot \pi\right)\right)}}{\pi}\right)} - 1\right) \]

7. Applied egg-rr 2.8%

   \[\leadsto -4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right)}{\pi}\right)} - 1\right)} \]
8. Step-by-step derivation

   1. expm1-def 2.8%

      \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \tanh \left(0.25 \cdot \left(f \cdot \pi\right)\right)}{\pi}\right)\right)} \]

   2. expm1-log1p 4.3%

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

9. Simplified 4.3%

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

10. Taylor expanded in f around 0 1.6%

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

    1. log-prod 1.6%

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

    2. associate-+r+ 1.6%

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

    3. log-prod 1.6%

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

    4. log-prod 1.6%

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

    5. associate-*l* 1.6%

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

12. Simplified 1.6%

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

13. Final simplification 1.6%

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

Alternative 5: 95.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

3. Taylor expanded in f around 0 94.5%

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

   1. *-commutative 94.5%

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

   2. associate-/r* 94.5%

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

   3. distribute-rgt-out-- 94.5%

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

   4. metadata-eval 94.5%

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

   5. associate-/r* 94.5%

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

5. Simplified 94.5%

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

6. Taylor expanded in f around 0 94.7%

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

   1. neg-mul-1 94.7%

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

   2. sub-neg 94.7%

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

8. Simplified 94.7%

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

9. Taylor expanded in f around inf 94.7%

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

    1. log-rec 94.7%

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

    2. mul-1-neg 94.7%

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

    3. remove-double-neg 94.7%

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

    4. log-div 94.6%

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

11. Simplified 94.6%

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

12. Final simplification 94.6%

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

Alternative 6: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log 8}{\pi} \cdot \left(--4\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ (log 8.0) PI) (- -4.0)))

C

double code(double f) {
	return (log(8.0) / ((double) M_PI)) * -(-4.0);
}

Java

public static double code(double f) {
	return (Math.log(8.0) / Math.PI) * -(-4.0);
}

Python

def code(f):
	return (math.log(8.0) / math.pi) * -(-4.0)

Julia

function code(f)
	return Float64(Float64(log(8.0) / pi) * Float64(-(-4.0)))
end

MATLAB

function tmp = code(f)
	tmp = (log(8.0) / pi) * -(-4.0);
end

Wolfram

code[f_] := N[(N[(N[Log[8.0], $MachinePrecision] / Pi), $MachinePrecision] * (--4.0)), $MachinePrecision]

Derivation

1. Initial program 6.4%

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

4. Applied egg-rr 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{16}}\right) \]
5. Step-by-step derivation

   1. clear-num 1.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{1}{\frac{16}{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}\right)} \]

   2. log-div 1.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log 1 - \log \left(\frac{16}{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}\right)\right)} \]

   3. metadata-eval 1.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\color{blue}{0} - \log \left(\frac{16}{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}\right)\right) \]

   4. cosh-undef 1.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0 - \log \left(\frac{16}{\color{blue}{2 \cdot \cosh \left(\frac{\pi}{4} \cdot f\right)}}\right)\right) \]

   5. div-inv 1.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0 - \log \left(\frac{16}{2 \cdot \cosh \left(\color{blue}{\left(\pi \cdot \frac{1}{4}\right)} \cdot f\right)}\right)\right) \]

   6. metadata-eval 1.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0 - \log \left(\frac{16}{2 \cdot \cosh \left(\left(\pi \cdot \color{blue}{0.25}\right) \cdot f\right)}\right)\right) \]

6. Applied egg-rr 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(0 - \log \left(\frac{16}{2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}\right)\right)} \]
7. Step-by-step derivation

   1. neg-sub0 1.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(-\log \left(\frac{16}{2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}\right)\right)} \]

   2. associate-/r* 1.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(-\log \color{blue}{\left(\frac{\frac{16}{2}}{\cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}\right)}\right) \]

   3. metadata-eval 1.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(-\log \left(\frac{\color{blue}{8}}{\cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}\right)\right) \]

   4. *-commutative 1.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(-\log \left(\frac{8}{\cosh \color{blue}{\left(f \cdot \left(\pi \cdot 0.25\right)\right)}}\right)\right) \]

8. Simplified 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(-\log \left(\frac{8}{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}\right)\right)} \]

9. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{-4 \cdot \frac{\log 8}{\pi}} \]

10. Final simplification 1.6%

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

Reproduce

herbie shell --seed 2024014 
(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)))))))))