VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.0% → 96.6%

Time: 22.2s

Alternatives: 5

Speedup: 4.8×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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, 2.5× speedup

Math

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

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(4.0 / Float64(f * pi)))) / Float64(pi * 0.25)))
end

Wolfram

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

Derivation

1. Initial program 6.2%

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

3. Taylor expanded in f around 0 95.4%

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

5. Simplified 95.4%

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

   1. associate-*l/ 95.5%

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

   2. *-un-lft-identity 95.5%

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

   3. associate-/r* 95.5%

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

   4. metadata-eval 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{\color{blue}{0.010416666666666666}}{\frac{0.5}{\pi}}, -2, 0.0625 \cdot \left(2 \cdot \pi\right)\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}} \]

   5. associate-*r* 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, \color{blue}{\left(0.0625 \cdot 2\right) \cdot \pi}\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}} \]

   6. metadata-eval 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, \color{blue}{0.125} \cdot \pi\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}} \]

   7. associate-/l/ 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, 0.125 \cdot \pi\right), \color{blue}{\frac{4}{f \cdot \pi}}\right)\right)}{\frac{\pi}{4}} \]

   8. div-inv 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, 0.125 \cdot \pi\right), \frac{4}{f \cdot \pi}\right)\right)}{\color{blue}{\pi \cdot \frac{1}{4}}} \]

   9. metadata-eval 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, 0.125 \cdot \pi\right), \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot \color{blue}{0.25}} \]

7. Applied egg-rr 95.5%

   \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, 0.125 \cdot \pi\right), \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25}} \]
8. Step-by-step derivation

   1. fma-def 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\frac{0.010416666666666666}{\frac{0.5}{\pi}} \cdot -2 + 0.125 \cdot \pi}, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   2. *-commutative 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{-2 \cdot \frac{0.010416666666666666}{\frac{0.5}{\pi}}} + 0.125 \cdot \pi, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   3. associate-/r/ 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, -2 \cdot \color{blue}{\left(\frac{0.010416666666666666}{0.5} \cdot \pi\right)} + 0.125 \cdot \pi, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   4. metadata-eval 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, -2 \cdot \left(\color{blue}{0.020833333333333332} \cdot \pi\right) + 0.125 \cdot \pi, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   5. associate-*r* 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\left(-2 \cdot 0.020833333333333332\right) \cdot \pi} + 0.125 \cdot \pi, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   6. metadata-eval 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{-0.041666666666666664} \cdot \pi + 0.125 \cdot \pi, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   7. distribute-rgt-out 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(-0.041666666666666664 + 0.125\right)}, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   8. metadata-eval 95.5%

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

   9. *-commutative 95.5%

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

9. Simplified 95.5%

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

10. Final simplification 95.5%

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

Alternative 2: 1.6% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.2%

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

3. Taylor expanded in f around 0 95.4%

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

5. Simplified 95.4%

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

   1. associate-*l/ 95.5%

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

   2. *-un-lft-identity 95.5%

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

   3. associate-/r* 95.5%

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

   4. metadata-eval 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{\color{blue}{0.010416666666666666}}{\frac{0.5}{\pi}}, -2, 0.0625 \cdot \left(2 \cdot \pi\right)\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}} \]

   5. associate-*r* 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, \color{blue}{\left(0.0625 \cdot 2\right) \cdot \pi}\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}} \]

   6. metadata-eval 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, \color{blue}{0.125} \cdot \pi\right), \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\pi}{4}} \]

   7. associate-/l/ 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, 0.125 \cdot \pi\right), \color{blue}{\frac{4}{f \cdot \pi}}\right)\right)}{\frac{\pi}{4}} \]

   8. div-inv 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, 0.125 \cdot \pi\right), \frac{4}{f \cdot \pi}\right)\right)}{\color{blue}{\pi \cdot \frac{1}{4}}} \]

   9. metadata-eval 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, 0.125 \cdot \pi\right), \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot \color{blue}{0.25}} \]

7. Applied egg-rr 95.5%

   \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, 0.125 \cdot \pi\right), \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25}} \]
8. Step-by-step derivation

   1. fma-def 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\frac{0.010416666666666666}{\frac{0.5}{\pi}} \cdot -2 + 0.125 \cdot \pi}, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   2. *-commutative 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{-2 \cdot \frac{0.010416666666666666}{\frac{0.5}{\pi}}} + 0.125 \cdot \pi, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   3. associate-/r/ 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, -2 \cdot \color{blue}{\left(\frac{0.010416666666666666}{0.5} \cdot \pi\right)} + 0.125 \cdot \pi, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   4. metadata-eval 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, -2 \cdot \left(\color{blue}{0.020833333333333332} \cdot \pi\right) + 0.125 \cdot \pi, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   5. associate-*r* 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\left(-2 \cdot 0.020833333333333332\right) \cdot \pi} + 0.125 \cdot \pi, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   6. metadata-eval 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{-0.041666666666666664} \cdot \pi + 0.125 \cdot \pi, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   7. distribute-rgt-out 95.5%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(-0.041666666666666664 + 0.125\right)}, \frac{4}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   8. metadata-eval 95.5%

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

   9. *-commutative 95.5%

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

9. Simplified 95.5%

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

10. Taylor expanded in f around inf 1.6%

    \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(0.08333333333333333 \cdot \pi\right) + -1 \cdot \log \left(\frac{1}{f}\right)}{\pi}} \]
11. Step-by-step derivation

    1. associate-*r/ 1.6%

       \[\leadsto -\color{blue}{\frac{4 \cdot \left(\log \left(0.08333333333333333 \cdot \pi\right) + -1 \cdot \log \left(\frac{1}{f}\right)\right)}{\pi}} \]

    2. *-rgt-identity 1.6%

       \[\leadsto -\frac{4 \cdot \left(\log \left(0.08333333333333333 \cdot \pi\right) + -1 \cdot \log \left(\frac{1}{f}\right)\right)}{\color{blue}{\pi \cdot 1}} \]

    3. times-frac 1.6%

       \[\leadsto -\color{blue}{\frac{4}{\pi} \cdot \frac{\log \left(0.08333333333333333 \cdot \pi\right) + -1 \cdot \log \left(\frac{1}{f}\right)}{1}} \]

    4. log-prod 1.6%

       \[\leadsto -\frac{4}{\pi} \cdot \frac{\color{blue}{\left(\log 0.08333333333333333 + \log \pi\right)} + -1 \cdot \log \left(\frac{1}{f}\right)}{1} \]

    5. +-commutative 1.6%

       \[\leadsto -\frac{4}{\pi} \cdot \frac{\color{blue}{\left(\log \pi + \log 0.08333333333333333\right)} + -1 \cdot \log \left(\frac{1}{f}\right)}{1} \]

    6. log-rec 1.6%

       \[\leadsto -\frac{4}{\pi} \cdot \frac{\left(\log \pi + \log 0.08333333333333333\right) + -1 \cdot \color{blue}{\left(-\log f\right)}}{1} \]

    7. mul-1-neg 1.6%

       \[\leadsto -\frac{4}{\pi} \cdot \frac{\left(\log \pi + \log 0.08333333333333333\right) + \color{blue}{\left(-\left(-\log f\right)\right)}}{1} \]

    8. remove-double-neg 1.6%

       \[\leadsto -\frac{4}{\pi} \cdot \frac{\left(\log \pi + \log 0.08333333333333333\right) + \color{blue}{\log f}}{1} \]

    9. associate-+l+ 1.6%

       \[\leadsto -\frac{4}{\pi} \cdot \frac{\color{blue}{\log \pi + \left(\log 0.08333333333333333 + \log f\right)}}{1} \]

    10. log-prod 1.6%

        \[\leadsto -\frac{4}{\pi} \cdot \frac{\log \pi + \color{blue}{\log \left(0.08333333333333333 \cdot f\right)}}{1} \]

    11. *-commutative 1.6%

        \[\leadsto -\frac{4}{\pi} \cdot \frac{\log \pi + \log \color{blue}{\left(f \cdot 0.08333333333333333\right)}}{1} \]

    12. log-prod 1.6%

        \[\leadsto -\frac{4}{\pi} \cdot \frac{\color{blue}{\log \left(\pi \cdot \left(f \cdot 0.08333333333333333\right)\right)}}{1} \]

    13. *-commutative 1.6%

        \[\leadsto -\frac{4}{\pi} \cdot \frac{\log \color{blue}{\left(\left(f \cdot 0.08333333333333333\right) \cdot \pi\right)}}{1} \]

    14. rem-square-sqrt 0.2%

        \[\leadsto -\frac{4}{\pi} \cdot \frac{\color{blue}{\sqrt{\log \left(\left(f \cdot 0.08333333333333333\right) \cdot \pi\right)} \cdot \sqrt{\log \left(\left(f \cdot 0.08333333333333333\right) \cdot \pi\right)}}}{1} \]

12. Simplified 1.6%

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

13. Taylor expanded in f around 0 1.6%

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

    1. *-commutative 1.6%

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

    2. *-commutative 1.6%

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

    3. *-commutative 1.6%

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

    4. associate-*l* 1.6%

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

15. Simplified 1.6%

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

16. Final simplification 1.6%

    \[\leadsto \log \left(f \cdot \left(\pi \cdot 0.08333333333333333\right)\right) \cdot \frac{-4}{\pi} \]
17. Add Preprocessing

Alternative 3: 95.8% accurate, 4.8× 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(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 6.2%

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

3. Taylor expanded in f around 0 95.2%

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

   1. associate-*r/ 95.2%

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

   2. associate-/l* 95.1%

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

   3. associate-/r/ 95.2%

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

   4. mul-1-neg 95.2%

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

   5. unsub-neg 95.2%

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

   6. distribute-rgt-out-- 95.2%

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

   7. *-commutative 95.2%

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

   8. associate-/r* 95.2%

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

   9. metadata-eval 95.2%

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

   10. metadata-eval 95.2%

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

5. Simplified 95.2%

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

   1. diff-log 95.1%

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

   2. associate-/l/ 95.1%

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

7. Applied egg-rr 95.1%

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

8. Final simplification 95.1%

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

Alternative 4: 95.9% accurate, 4.8× speedup

Math

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

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

Derivation

1. Initial program 6.2%

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

3. Taylor expanded in f around 0 95.1%

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

   1. associate-/l/ 95.1%

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

   2. distribute-rgt-out-- 95.1%

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

   3. *-commutative 95.1%

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

   4. associate-/r* 95.1%

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

   5. metadata-eval 95.1%

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

   6. metadata-eval 95.1%

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

5. Simplified 95.1%

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

   1. diff-log 95.2%

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

   2. clear-num 95.2%

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

   3. associate-*l/ 95.2%

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

   4. diff-log 95.2%

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

   5. associate-/l/ 95.2%

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

7. Applied egg-rr 95.2%

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

8. Final simplification 95.2%

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

Alternative 5: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \log 0.125 \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (log 0.125) (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(0.125) * (-1.0 / (((double) M_PI) / 4.0));
}

Java

public static double code(double f) {
	return Math.log(0.125) * (-1.0 / (Math.PI / 4.0));
}

Python

def code(f):
	return math.log(0.125) * (-1.0 / (math.pi / 4.0))

Julia

function code(f)
	return Float64(log(0.125) * Float64(-1.0 / Float64(pi / 4.0)))
end

MATLAB

function tmp = code(f)
	tmp = log(0.125) * (-1.0 / (pi / 4.0));
end

Wolfram

code[f_] := N[(N[Log[0.125], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.2%

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

4. Applied egg-rr 1.6%

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

5. Taylor expanded in f around 0 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\log 0.125} \]

6. Final simplification 1.6%

   \[\leadsto \log 0.125 \cdot \frac{-1}{\frac{\pi}{4}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(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)))))))))