VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 98.9%

Time: 33.1s

Alternatives: 7

Speedup: 4.9×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. distribute-lft-neg-in 7.2%

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

   2. distribute-neg-frac2 7.2%

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

   3. associate-*l/ 7.1%

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

   4. *-lft-identity 7.1%

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

3. Simplified 7.1%

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

5. Taylor expanded in f around inf 7.1%

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

7. Simplified 99.2%

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

Alternative 2: 98.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. distribute-lft-neg-in 7.2%

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

   2. distribute-neg-frac2 7.2%

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

   3. associate-*l/ 7.1%

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

   4. *-lft-identity 7.1%

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

3. Simplified 7.1%

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

5. Taylor expanded in f around inf 7.1%

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

7. Simplified 99.2%

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

8. Final simplification 99.2%

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

Alternative 3: 98.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. distribute-lft-neg-in 7.2%

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

   2. distribute-neg-frac2 7.2%

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

   3. associate-*l/ 7.1%

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

   4. *-lft-identity 7.1%

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

3. Simplified 7.1%

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

5. Taylor expanded in f around inf 7.1%

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

7. Simplified 99.2%

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

   1. frac-2neg 99.2%

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

   2. metadata-eval 99.2%

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

   3. *-commutative 99.2%

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

   4. *-commutative 99.2%

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

   5. neg-log 99.2%

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

   6. div-inv 99.2%

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

   7. metadata-eval 99.2%

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

   8. distribute-lft-neg-in 99.2%

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

   9. frac-2neg 99.2%

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

   10. *-commutative 99.2%

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

   11. *-un-lft-identity 99.2%

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

   12. times-frac 99.2%

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

   13. metadata-eval 99.2%

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

9. Applied egg-rr 99.2%

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

    1. associate-*r/ 99.2%

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

    2. *-rgt-identity 99.2%

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

    3. times-frac 99.1%

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

    4. /-rgt-identity 99.1%

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

11. Simplified 99.1%

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

Alternative 4: 96.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around 0 96.8%

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

   1. mul-1-neg 96.8%

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

   2. unsub-neg 96.8%

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

7. Simplified 96.8%

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

Alternative 5: 96.3% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. distribute-lft-neg-in 7.2%

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

   2. distribute-neg-frac2 7.2%

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

   3. associate-*l/ 7.1%

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

   4. *-lft-identity 7.1%

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

3. Simplified 7.1%

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

5. Taylor expanded in f around inf 7.1%

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

7. Simplified 99.2%

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

8. Taylor expanded in f around 0 96.7%

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

   1. *-commutative 96.7%

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

10. Simplified 96.7%

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

11. Final simplification 96.7%

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

Alternative 6: 96.1% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. distribute-lft-neg-in 7.2%

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

   2. distribute-neg-frac2 7.2%

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

   3. associate-*l/ 7.1%

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

   4. *-lft-identity 7.1%

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

3. Simplified 7.1%

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

5. Taylor expanded in f around inf 7.1%

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

7. Simplified 99.2%

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

8. Taylor expanded in f around 0 96.7%

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

   1. *-commutative 96.7%

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

10. Simplified 96.7%

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

    1. div-inv 96.6%

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

    2. clear-num 96.6%

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

12. Applied egg-rr 96.6%

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

13. Final simplification 96.6%

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

Alternative 7: 0.7% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. distribute-lft-neg-in 7.2%

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

   2. distribute-neg-frac2 7.2%

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

   3. associate-*l/ 7.1%

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

   4. *-lft-identity 7.1%

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

3. Simplified 7.1%

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

5. Taylor expanded in f around inf 7.1%

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

7. Simplified 99.2%

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

9. Applied egg-rr 0.6%

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

    1. +-inverses 0.7%

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

11. Simplified 0.7%

    \[\leadsto \frac{\log \color{blue}{0}}{\frac{\pi}{-4}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024105 
(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)))))))))