VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.0% → 96.6%

Time: 26.3s

Alternatives: 5

Speedup: 3.3×

• could not determine a ground truth

  1. f = -8.46718225391735e+87

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{\frac{4}{f}}{\pi}\right)\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (- (/ (log (fma f (* PI 0.08333333333333333) (/ (/ 4.0 f) PI))) (* PI 0.25))))

C

double code(double f) {
	return -(log(fma(f, (((double) M_PI) * 0.08333333333333333), ((4.0 / f) / ((double) M_PI)))) / (((double) M_PI) * 0.25));
}

Julia

function code(f)
	return Float64(-Float64(log(fma(f, Float64(pi * 0.08333333333333333), Float64(Float64(4.0 / f) / pi))) / Float64(pi * 0.25)))
end

Wolfram

code[f_] := (-N[(N[Log[N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(N[(4.0 / f), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision])

Derivation

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. Taylor expanded in f around 0 97.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(2 \cdot \frac{1}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f} + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right)\right)\right)\right)} \]

4. Simplified 97.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), \frac{\frac{4}{\pi}}{f}\right)\right)} \]
5. Step-by-step derivation

   1. associate-*l/ 97.1%

      \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}}} \]

   2. *-un-lft-identity 97.1%

      \[\leadsto -\frac{\color{blue}{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), \frac{\frac{4}{\pi}}{f}\right)\right)}}{\frac{\pi}{4}} \]

   3. associate-*r/ 97.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\color{blue}{\frac{0.5 \cdot 0.5}{\pi}}}, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}} \]

   4. metadata-eval 97.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{\color{blue}{0.25}}{\pi}}, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}} \]

   5. div-inv 97.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, 0.0625 \cdot \color{blue}{\left(\pi \cdot \frac{1}{0.5}\right)}\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}} \]

   6. metadata-eval 97.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, 0.0625 \cdot \left(\pi \cdot \color{blue}{2}\right)\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}} \]

   7. associate-/l/ 97.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, 0.0625 \cdot \left(\pi \cdot 2\right)\right), \color{blue}{\frac{4}{f \cdot \pi}}\right)\right)}{\frac{\pi}{4}} \]

   8. *-commutative 97.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, 0.0625 \cdot \left(\pi \cdot 2\right)\right), \frac{4}{\color{blue}{\pi \cdot f}}\right)\right)}{\frac{\pi}{4}} \]

   9. div-inv 97.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, 0.0625 \cdot \left(\pi \cdot 2\right)\right), \frac{4}{\pi \cdot f}\right)\right)}{\color{blue}{\pi \cdot \frac{1}{4}}} \]

   10. metadata-eval 97.1%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, 0.0625 \cdot \left(\pi \cdot 2\right)\right), \frac{4}{\pi \cdot f}\right)\right)}{\pi \cdot \color{blue}{0.25}} \]

6. Applied egg-rr 97.1%

   \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, 0.0625 \cdot \left(\pi \cdot 2\right)\right), \frac{4}{\pi \cdot f}\right)\right)}{\pi \cdot 0.25}} \]
7. Step-by-step derivation

8. Simplified 97.1%

   \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{\frac{4}{f}}{\pi}\right)\right)}{\pi \cdot 0.25}} \]

9. Final simplification 97.1%

   \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{\frac{4}{f}}{\pi}\right)\right)}{\pi \cdot 0.25} \]

Alternative 2: 96.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(f \cdot \left(\pi \cdot 0.08333333333333333\right) + \frac{4}{f \cdot \pi}\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (- (log (+ (* f (* PI 0.08333333333333333)) (/ 4.0 (* f PI)))))
  (* PI 0.25)))

C

double code(double f) {
	return -log(((f * (((double) M_PI) * 0.08333333333333333)) + (4.0 / (f * ((double) M_PI))))) / (((double) M_PI) * 0.25);
}

Java

public static double code(double f) {
	return -Math.log(((f * (Math.PI * 0.08333333333333333)) + (4.0 / (f * Math.PI)))) / (Math.PI * 0.25);
}

Python

def code(f):
	return -math.log(((f * (math.pi * 0.08333333333333333)) + (4.0 / (f * math.pi)))) / (math.pi * 0.25)

Julia

function code(f)
	return Float64(Float64(-log(Float64(Float64(f * Float64(pi * 0.08333333333333333)) + Float64(4.0 / Float64(f * pi))))) / Float64(pi * 0.25))
end

MATLAB

function tmp = code(f)
	tmp = -log(((f * (pi * 0.08333333333333333)) + (4.0 / (f * pi)))) / (pi * 0.25);
end

Wolfram

code[f_] := N[((-N[Log[N[(N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + N[(4.0 / N[(f * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

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. Taylor expanded in f around 0 97.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(2 \cdot \frac{1}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f} + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right)\right)\right)\right)} \]

4. Simplified 97.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), \frac{\frac{4}{\pi}}{f}\right)\right)} \]
5. Step-by-step derivation

   1. associate-*l/ 97.1%

      \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}}} \]

   2. *-un-lft-identity 97.1%

      \[\leadsto -\frac{\color{blue}{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), \frac{\frac{4}{\pi}}{f}\right)\right)}}{\frac{\pi}{4}} \]

   3. associate-*r/ 97.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\color{blue}{\frac{0.5 \cdot 0.5}{\pi}}}, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}} \]

   4. metadata-eval 97.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{\color{blue}{0.25}}{\pi}}, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}} \]

   5. div-inv 97.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, 0.0625 \cdot \color{blue}{\left(\pi \cdot \frac{1}{0.5}\right)}\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}} \]

   6. metadata-eval 97.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, 0.0625 \cdot \left(\pi \cdot \color{blue}{2}\right)\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}} \]

   7. associate-/l/ 97.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, 0.0625 \cdot \left(\pi \cdot 2\right)\right), \color{blue}{\frac{4}{f \cdot \pi}}\right)\right)}{\frac{\pi}{4}} \]

   8. *-commutative 97.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, 0.0625 \cdot \left(\pi \cdot 2\right)\right), \frac{4}{\color{blue}{\pi \cdot f}}\right)\right)}{\frac{\pi}{4}} \]

   9. div-inv 97.1%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, 0.0625 \cdot \left(\pi \cdot 2\right)\right), \frac{4}{\pi \cdot f}\right)\right)}{\color{blue}{\pi \cdot \frac{1}{4}}} \]

   10. metadata-eval 97.1%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, 0.0625 \cdot \left(\pi \cdot 2\right)\right), \frac{4}{\pi \cdot f}\right)\right)}{\pi \cdot \color{blue}{0.25}} \]

6. Applied egg-rr 97.1%

   \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{\frac{0.25}{\pi}}, -2, 0.0625 \cdot \left(\pi \cdot 2\right)\right), \frac{4}{\pi \cdot f}\right)\right)}{\pi \cdot 0.25}} \]
7. Step-by-step derivation

8. Simplified 97.1%

   \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{\frac{4}{f}}{\pi}\right)\right)}{\pi \cdot 0.25}} \]
9. Step-by-step derivation

   1. fma-udef 97.1%

      \[\leadsto -\frac{\log \color{blue}{\left(f \cdot \left(\pi \cdot 0.08333333333333333\right) + \frac{\frac{4}{f}}{\pi}\right)}}{\pi \cdot 0.25} \]

   2. associate-/l/ 97.1%

      \[\leadsto -\frac{\log \left(f \cdot \left(\pi \cdot 0.08333333333333333\right) + \color{blue}{\frac{4}{\pi \cdot f}}\right)}{\pi \cdot 0.25} \]

10. Applied egg-rr 97.1%

    \[\leadsto -\frac{\log \color{blue}{\left(f \cdot \left(\pi \cdot 0.08333333333333333\right) + \frac{4}{\pi \cdot f}\right)}}{\pi \cdot 0.25} \]

11. Final simplification 97.1%

    \[\leadsto \frac{-\log \left(f \cdot \left(\pi \cdot 0.08333333333333333\right) + \frac{4}{f \cdot \pi}\right)}{\pi \cdot 0.25} \]

Alternative 3: 96.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. *-commutative 6.6%

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

   2. distribute-rgt-neg-in 6.6%

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

3. Simplified 6.6%

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

4. Taylor expanded in f around 0 96.7%

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

   1. *-commutative 96.7%

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

   2. mul-1-neg 96.7%

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

   3. unsub-neg 96.7%

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

   4. distribute-rgt-out-- 96.7%

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

   5. *-commutative 96.7%

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

   6. associate-/r* 96.7%

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

   7. metadata-eval 96.7%

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

   8. metadata-eval 96.7%

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

6. Simplified 96.7%

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

7. Final simplification 96.7%

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

Alternative 4: 96.1% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. *-commutative 6.6%

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

   2. distribute-rgt-neg-in 6.6%

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

3. Simplified 6.6%

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

4. Taylor expanded in f around -inf 6.6%

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

5. Taylor expanded in f around 0 97.1%

   \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\color{blue}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}}\right)}{\pi} \]
6. Step-by-step derivation

   1. *-commutative 97.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)} + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right)}{\pi} \]

   2. fma-def 97.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\color{blue}{\mathsf{fma}\left(f, 0.25 \cdot \pi - -0.25 \cdot \pi, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}}\right)}{\pi} \]

   3. distribute-rgt-out-- 97.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right)}{\pi} \]

   4. metadata-eval 97.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right)}{\pi} \]

   5. distribute-rgt-out-- 97.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, {f}^{3} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right)}\right)}{\pi} \]

   6. metadata-eval 97.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, {f}^{3} \cdot \left({\pi}^{3} \cdot \color{blue}{0.005208333333333333}\right)\right)}\right)}{\pi} \]

   7. associate-*r* 97.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{3} \cdot {\pi}^{3}\right) \cdot 0.005208333333333333}\right)}\right)}{\pi} \]

7. Simplified 97.1%

   \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{0.25 \cdot \left(f \cdot \pi\right)} + e^{-0.25 \cdot \left(f \cdot \pi\right)}}{\color{blue}{\mathsf{fma}\left(f, \pi \cdot 0.5, {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333\right)}}\right)}{\pi} \]

8. Taylor expanded in f around 0 96.7%

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

   1. mul-1-neg 96.7%

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

   2. log-rec 96.7%

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

   3. +-commutative 96.7%

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

   4. log-rec 96.7%

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

   5. unsub-neg 96.7%

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

   6. log-div 96.7%

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

   7. associate-/r* 96.7%

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

   8. *-commutative 96.7%

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

10. Simplified 96.7%

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

11. Final simplification 96.7%

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

Alternative 5: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \log 0.0078125 \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (log 0.0078125) (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(0.0078125) * (-1.0 / (((double) M_PI) / 4.0));
}

Java

public static double code(double f) {
	return Math.log(0.0078125) * (-1.0 / (Math.PI / 4.0));
}

Python

def code(f):
	return math.log(0.0078125) * (-1.0 / (math.pi / 4.0))

Julia

function code(f)
	return Float64(log(0.0078125) * Float64(-1.0 / Float64(pi / 4.0)))
end

MATLAB

function tmp = code(f)
	tmp = log(0.0078125) * (-1.0 / (pi / 4.0));
end

Wolfram

code[f_] := N[(N[Log[0.0078125], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

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

3. Applied egg-rr 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{256}}\right) \]

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\log 0.0078125} \]

5. Final simplification 1.6%

   \[\leadsto \log 0.0078125 \cdot \frac{-1}{\frac{\pi}{4}} \]

Reproduce

herbie shell --seed 2023273 
(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)))))))))