VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 97.3%

Time: 33.1s

Alternatives: 5

Speedup: 3.3×

• could not determine a ground truth

  1. f = -2.070484259355209e+245

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: 6.7% 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: 97.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.6%

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

   1. expm1-log1p-u 5.6%

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

   2. expm1-udef 5.6%

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

3. Applied egg-rr 97.4%

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

   1. expm1-def 97.3%

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

   2. expm1-log1p 98.6%

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

   3. *-lft-identity 98.6%

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

   4. *-commutative 98.6%

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

   5. times-frac 98.6%

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

   6. metadata-eval 98.6%

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

5. Simplified 98.6%

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

   1. clear-num 98.6%

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

   2. log-rec 98.6%

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

   3. div-inv 98.6%

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

   4. metadata-eval 98.6%

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

   5. div-inv 98.6%

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

   6. metadata-eval 98.6%

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

7. Applied egg-rr 98.6%

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

8. Final simplification 98.6%

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

Alternative 2: 97.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.6%

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

   1. expm1-log1p-u 5.6%

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

   2. expm1-udef 5.6%

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

3. Applied egg-rr 97.4%

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

   1. expm1-def 97.3%

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

   2. expm1-log1p 98.6%

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

   3. *-lft-identity 98.6%

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

   4. *-commutative 98.6%

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

   5. times-frac 98.6%

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

   6. metadata-eval 98.6%

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

5. Simplified 98.6%

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

6. Final simplification 98.6%

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

Alternative 3: 96.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.6%

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

2. Taylor expanded in f around 0 97.6%

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

   1. distribute-rgt-out-- 97.6%

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

   2. metadata-eval 97.6%

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

4. Simplified 97.6%

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

   1. add-sqr-sqrt 97.2%

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

   2. sqrt-unprod 97.7%

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

   3. pow2 97.7%

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

   4. associate-*l/ 97.8%

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

   5. div-inv 97.8%

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

   6. metadata-eval 97.8%

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

   7. *-commutative 97.8%

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

   8. times-frac 97.8%

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

   9. metadata-eval 97.8%

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

   10. associate-/r* 97.8%

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

6. Applied egg-rr 97.8%

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

   1. unpow2 97.8%

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

   2. rem-sqrt-square 97.8%

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

   3. log-div 97.8%

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

   4. log-div 97.8%

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

   5. *-commutative 97.8%

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

   6. associate--r+ 97.8%

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

8. Simplified 97.8%

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

9. Final simplification 97.8%

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

Alternative 4: 96.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.6%

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

   1. expm1-log1p-u 5.6%

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

   2. expm1-udef 5.6%

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

3. Applied egg-rr 97.4%

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

   1. expm1-def 97.3%

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

   2. expm1-log1p 98.6%

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

   3. *-lft-identity 98.6%

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

   4. *-commutative 98.6%

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

   5. times-frac 98.6%

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

   6. metadata-eval 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in f around 0 97.7%

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

   1. mul-1-neg 97.7%

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

   2. unsub-neg 97.7%

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

8. Simplified 97.7%

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

9. Final simplification 97.7%

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

Alternative 5: 96.1% accurate, 3.3× speedup

Math

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

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(4.0 / Float64(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[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.6%

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

   1. expm1-log1p-u 5.6%

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

   2. expm1-udef 5.6%

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

3. Applied egg-rr 97.4%

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

   1. expm1-def 97.3%

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

   2. expm1-log1p 98.6%

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

   3. *-lft-identity 98.6%

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

   4. *-commutative 98.6%

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

   5. times-frac 98.6%

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

   6. metadata-eval 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in f around 0 97.7%

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

   1. *-commutative 97.7%

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

8. Simplified 97.7%

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

9. Final simplification 97.7%

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

Reproduce

herbie shell --seed 2023187 
(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)))))))))