VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 97.2%

Time: 29.2s

Alternatives: 5

Speedup: 3.3×

• could not determine a ground truth

  1. f = -7.699920411964023e+145

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: 97.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{4}{f}}\\ -\frac{\log \left(\frac{\cosh t_0}{\sinh t_0}\right)}{\pi \cdot 0.25} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ PI (/ 4.0 f))))
   (- (/ (log (/ (cosh t_0) (sinh t_0))) (* PI 0.25)))))

C

double code(double f) {
	double t_0 = ((double) M_PI) / (4.0 / f);
	return -(log((cosh(t_0) / sinh(t_0))) / (((double) M_PI) * 0.25));
}

Java

public static double code(double f) {
	double t_0 = Math.PI / (4.0 / f);
	return -(Math.log((Math.cosh(t_0) / Math.sinh(t_0))) / (Math.PI * 0.25));
}

Python

def code(f):
	t_0 = math.pi / (4.0 / f)
	return -(math.log((math.cosh(t_0) / math.sinh(t_0))) / (math.pi * 0.25))

Julia

function code(f)
	t_0 = Float64(pi / Float64(4.0 / f))
	return Float64(-Float64(log(Float64(cosh(t_0) / sinh(t_0))) / Float64(pi * 0.25)))
end

MATLAB

function tmp = code(f)
	t_0 = pi / (4.0 / f);
	tmp = -(log((cosh(t_0) / sinh(t_0))) / (pi * 0.25));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(Pi / N[(4.0 / f), $MachinePrecision]), $MachinePrecision]}, (-N[(N[Log[N[(N[Cosh[t$95$0], $MachinePrecision] / N[Sinh[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Initial program 8.3%

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

   1. expm1-log1p-u 8.3%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\right)\right)} \]

   2. expm1-udef 8.3%

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

3. Applied egg-rr 97.5%

   \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{2 \cdot \sinh \left(\frac{\pi \cdot f}{4}\right)}\right)}{\pi \cdot 0.25}\right)} - 1\right)} \]
4. Step-by-step derivation

   1. expm1-def 97.5%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{2 \cdot \sinh \left(\frac{\pi \cdot f}{4}\right)}\right)}{\pi \cdot 0.25}\right)\right)} \]

   2. expm1-log1p 98.7%

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

   3. *-commutative 98.7%

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

   4. times-frac 98.7%

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

   5. metadata-eval 98.7%

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

   6. *-lft-identity 98.7%

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

   7. associate-/l* 98.7%

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

   8. associate-/l* 98.7%

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

5. Simplified 98.7%

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

6. Final simplification 98.7%

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

Alternative 2: 96.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 1 - \left(1 + \frac{\log \left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right)}{\pi \cdot 0.25}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (- 1.0 (+ 1.0 (/ (log (/ 2.0 (* f (* PI 0.5)))) (* PI 0.25)))))

C

double code(double f) {
	return 1.0 - (1.0 + (log((2.0 / (f * (((double) M_PI) * 0.5)))) / (((double) M_PI) * 0.25)));
}

Java

public static double code(double f) {
	return 1.0 - (1.0 + (Math.log((2.0 / (f * (Math.PI * 0.5)))) / (Math.PI * 0.25)));
}

Python

def code(f):
	return 1.0 - (1.0 + (math.log((2.0 / (f * (math.pi * 0.5)))) / (math.pi * 0.25)))

Julia

function code(f)
	return Float64(1.0 - Float64(1.0 + Float64(log(Float64(2.0 / Float64(f * Float64(pi * 0.5)))) / Float64(pi * 0.25))))
end

MATLAB

function tmp = code(f)
	tmp = 1.0 - (1.0 + (log((2.0 / (f * (pi * 0.5)))) / (pi * 0.25)));
end

Wolfram

code[f_] := N[(1.0 - N[(1.0 + N[(N[Log[N[(2.0 / N[(f * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.3%

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

2. Taylor expanded in f around 0 96.9%

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

   1. distribute-rgt-out-- 96.9%

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

   2. metadata-eval 96.9%

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

4. Simplified 96.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)} \]
5. Step-by-step derivation

   1. expm1-log1p-u 95.9%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)\right)\right)} \]

   2. associate-*l/ 95.9%

      \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \log \left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)}{\frac{\pi}{4}}}\right)\right) \]

   3. *-un-lft-identity 95.9%

      \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\log \left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)}}{\frac{\pi}{4}}\right)\right) \]

   4. associate-/r* 95.9%

      \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \color{blue}{\left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)}}{\frac{\pi}{4}}\right)\right) \]

   5. associate-/r* 95.9%

      \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\color{blue}{\frac{\frac{2}{\pi}}{0.5}}}{f}\right)}{\frac{\pi}{4}}\right)\right) \]

   6. div-inv 95.9%

      \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)}{\color{blue}{\pi \cdot \frac{1}{4}}}\right)\right) \]

   7. metadata-eval 95.9%

      \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)}{\pi \cdot \color{blue}{0.25}}\right)\right) \]

6. Applied egg-rr 95.9%

   \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)}{\pi \cdot 0.25}\right)\right)} \]
7. Step-by-step derivation

   1. expm1-udef 95.9%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)}{\pi \cdot 0.25}\right)} - 1\right)} \]

   2. log1p-udef 95.9%

      \[\leadsto -\left(e^{\color{blue}{\log \left(1 + \frac{\log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)}{\pi \cdot 0.25}\right)}} - 1\right) \]

   3. add-exp-log 97.1%

      \[\leadsto -\left(\color{blue}{\left(1 + \frac{\log \left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)}{\pi \cdot 0.25}\right)} - 1\right) \]

   4. div-inv 97.1%

      \[\leadsto -\left(\left(1 + \frac{\log \color{blue}{\left(\frac{\frac{2}{\pi}}{0.5} \cdot \frac{1}{f}\right)}}{\pi \cdot 0.25}\right) - 1\right) \]

   5. associate-/l/ 97.1%

      \[\leadsto -\left(\left(1 + \frac{\log \left(\color{blue}{\frac{2}{0.5 \cdot \pi}} \cdot \frac{1}{f}\right)}{\pi \cdot 0.25}\right) - 1\right) \]

   6. frac-times 97.1%

      \[\leadsto -\left(\left(1 + \frac{\log \color{blue}{\left(\frac{2 \cdot 1}{\left(0.5 \cdot \pi\right) \cdot f}\right)}}{\pi \cdot 0.25}\right) - 1\right) \]

   7. metadata-eval 97.1%

      \[\leadsto -\left(\left(1 + \frac{\log \left(\frac{\color{blue}{2}}{\left(0.5 \cdot \pi\right) \cdot f}\right)}{\pi \cdot 0.25}\right) - 1\right) \]

8. Applied egg-rr 97.1%

   \[\leadsto -\color{blue}{\left(\left(1 + \frac{\log \left(\frac{2}{\left(0.5 \cdot \pi\right) \cdot f}\right)}{\pi \cdot 0.25}\right) - 1\right)} \]

9. Final simplification 97.1%

   \[\leadsto 1 - \left(1 + \frac{\log \left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right)}{\pi \cdot 0.25}\right) \]

Alternative 3: 95.9% accurate, 3.3× 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 8.3%

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

   1. expm1-log1p-u 8.3%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\right)\right)} \]

   2. expm1-udef 8.3%

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

3. Applied egg-rr 97.5%

   \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{2 \cdot \sinh \left(\frac{\pi \cdot f}{4}\right)}\right)}{\pi \cdot 0.25}\right)} - 1\right)} \]
4. Step-by-step derivation

   1. expm1-def 97.5%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{2 \cdot \sinh \left(\frac{\pi \cdot f}{4}\right)}\right)}{\pi \cdot 0.25}\right)\right)} \]

   2. expm1-log1p 98.7%

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

   3. *-commutative 98.7%

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

   4. times-frac 98.7%

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

   5. metadata-eval 98.7%

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

   6. *-lft-identity 98.7%

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

   7. associate-/l* 98.7%

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

   8. associate-/l* 98.7%

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

5. Simplified 98.7%

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

6. Taylor expanded in f around 0 97.1%

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

   1. *-commutative 97.1%

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

   2. expm1-log1p-u 95.9%

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

   3. expm1-udef 95.9%

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

   4. sub-neg 95.9%

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

8. Applied egg-rr 76.7%

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

   1. +-commutative 76.7%

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

   2. associate-+r+ 76.7%

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

   3. metadata-eval 76.7%

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

   4. metadata-eval 76.7%

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

   5. log-pow 96.9%

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

   6. metadata-eval 96.9%

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

   7. associate-/r* 96.9%

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

   8. *-commutative 96.9%

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

   9. associate-/r/ 97.0%

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

   10. metadata-eval 97.0%

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

   11. associate-/r* 97.0%

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

   12. neg-mul-1 97.0%

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

   13. *-commutative 97.0%

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

   14. associate-/l* 97.0%

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

   15. distribute-neg-frac 97.0%

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

   16. metadata-eval 97.0%

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

10. Simplified 96.9%

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

11. Final simplification 96.9%

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

Alternative 4: 96.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

2. Taylor expanded in f around 0 97.0%

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

   1. associate-*r/ 97.0%

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

   2. associate-/l* 96.9%

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

   3. mul-1-neg 96.9%

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

   4. unsub-neg 96.9%

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

   5. distribute-rgt-out-- 96.9%

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

   6. metadata-eval 96.9%

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

4. Simplified 96.9%

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

5. Taylor expanded in f around 0 96.9%

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

   1. metadata-eval 96.9%

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

   2. associate-/r* 96.9%

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

   3. associate-/l/ 96.9%

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

   4. log-div 97.0%

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

   5. associate-/l/ 97.0%

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

   6. associate-/r* 97.0%

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

   7. metadata-eval 97.0%

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

   8. associate-/r* 97.0%

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

   9. *-commutative 97.0%

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

7. Simplified 97.0%

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

8. Final simplification 97.0%

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

Alternative 5: 96.1% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

   1. expm1-log1p-u 8.3%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\right)\right)} \]

   2. expm1-udef 8.3%

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

3. Applied egg-rr 97.5%

   \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{2 \cdot \sinh \left(\frac{\pi \cdot f}{4}\right)}\right)}{\pi \cdot 0.25}\right)} - 1\right)} \]
4. Step-by-step derivation

   1. expm1-def 97.5%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{2 \cdot \sinh \left(\frac{\pi \cdot f}{4}\right)}\right)}{\pi \cdot 0.25}\right)\right)} \]

   2. expm1-log1p 98.7%

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

   3. *-commutative 98.7%

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

   4. times-frac 98.7%

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

   5. metadata-eval 98.7%

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

   6. *-lft-identity 98.7%

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

   7. associate-/l* 98.7%

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

   8. associate-/l* 98.7%

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

5. Simplified 98.7%

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

6. Taylor expanded in f around 0 97.1%

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

7. Final simplification 97.1%

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

Reproduce

herbie shell --seed 2023249 
(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)))))))))