VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 96.3%

Time: 21.4s

Alternatives: 7

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: 6.9% 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: 96.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \log \left(\sqrt[3]{\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, -0.16666666666666666\right), \frac{4}{\pi \cdot f}\right)}\right) \cdot \frac{-12}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log (cbrt (fma f (fma PI 0.125 -0.16666666666666666) (/ 4.0 (* PI f)))))
  (/ (- 12.0) PI)))

C

double code(double f) {
	return log(cbrt(fma(f, fma(((double) M_PI), 0.125, -0.16666666666666666), (4.0 / (((double) M_PI) * f))))) * (-12.0 / ((double) M_PI));
}

Julia

function code(f)
	return Float64(log(cbrt(fma(f, fma(pi, 0.125, -0.16666666666666666), Float64(4.0 / Float64(pi * f))))) * Float64(Float64(-12.0) / pi))
end

Wolfram

code[f_] := N[(N[Log[N[Power[N[(f * N[(Pi * 0.125 + -0.16666666666666666), $MachinePrecision] + N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * N[((-12.0) / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.1%

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

3. Taylor expanded in f around 0 95.2%

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

5. Simplified 95.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), \frac{\frac{2}{f}}{\pi \cdot 0.5}\right)\right)} \]

7. Applied egg-rr 95.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\color{blue}{4} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), \frac{\frac{2}{f}}{\pi \cdot 0.5}\right)\right) \]
8. Step-by-step derivation

   1. add-cube-cbrt 95.1%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\left(\sqrt[3]{\mathsf{fma}\left(f, \mathsf{fma}\left(4 \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), \frac{\frac{2}{f}}{\pi \cdot 0.5}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(f, \mathsf{fma}\left(4 \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), \frac{\frac{2}{f}}{\pi \cdot 0.5}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(f, \mathsf{fma}\left(4 \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), \frac{\frac{2}{f}}{\pi \cdot 0.5}\right)}\right)} \]

   2. pow3 95.1%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left({\left(\sqrt[3]{\mathsf{fma}\left(f, \mathsf{fma}\left(4 \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), \frac{\frac{2}{f}}{\pi \cdot 0.5}\right)}\right)}^{3}\right)} \]

9. Applied egg-rr 95.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left({\left(\sqrt[3]{\mathsf{fma}\left(f, -0.16666666666666666 + \left(\pi \cdot 0.0625\right) \cdot 2, 4 \cdot \frac{\frac{1}{f}}{\pi}\right)}\right)}^{3}\right)} \]
10. Step-by-step derivation

    1. associate-*l/ 95.3%

       \[\leadsto -\color{blue}{\frac{1 \cdot \log \left({\left(\sqrt[3]{\mathsf{fma}\left(f, -0.16666666666666666 + \left(\pi \cdot 0.0625\right) \cdot 2, 4 \cdot \frac{\frac{1}{f}}{\pi}\right)}\right)}^{3}\right)}{\frac{\pi}{4}}} \]

    2. associate-/l* 95.1%

       \[\leadsto -\color{blue}{\frac{1}{\frac{\frac{\pi}{4}}{\log \left({\left(\sqrt[3]{\mathsf{fma}\left(f, -0.16666666666666666 + \left(\pi \cdot 0.0625\right) \cdot 2, 4 \cdot \frac{\frac{1}{f}}{\pi}\right)}\right)}^{3}\right)}}} \]

    3. clear-num 95.3%

       \[\leadsto -\color{blue}{\frac{\log \left({\left(\sqrt[3]{\mathsf{fma}\left(f, -0.16666666666666666 + \left(\pi \cdot 0.0625\right) \cdot 2, 4 \cdot \frac{\frac{1}{f}}{\pi}\right)}\right)}^{3}\right)}{\frac{\pi}{4}}} \]

    4. log-pow 95.1%

       \[\leadsto -\frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(f, -0.16666666666666666 + \left(\pi \cdot 0.0625\right) \cdot 2, 4 \cdot \frac{\frac{1}{f}}{\pi}\right)}\right)}}{\frac{\pi}{4}} \]

    5. associate-/l* 95.1%

       \[\leadsto -\color{blue}{\frac{3}{\frac{\frac{\pi}{4}}{\log \left(\sqrt[3]{\mathsf{fma}\left(f, -0.16666666666666666 + \left(\pi \cdot 0.0625\right) \cdot 2, 4 \cdot \frac{\frac{1}{f}}{\pi}\right)}\right)}}} \]

    6. div-inv 95.1%

       \[\leadsto -\frac{3}{\frac{\color{blue}{\pi \cdot \frac{1}{4}}}{\log \left(\sqrt[3]{\mathsf{fma}\left(f, -0.16666666666666666 + \left(\pi \cdot 0.0625\right) \cdot 2, 4 \cdot \frac{\frac{1}{f}}{\pi}\right)}\right)}} \]

    7. metadata-eval 95.1%

       \[\leadsto -\frac{3}{\frac{\pi \cdot \color{blue}{0.25}}{\log \left(\sqrt[3]{\mathsf{fma}\left(f, -0.16666666666666666 + \left(\pi \cdot 0.0625\right) \cdot 2, 4 \cdot \frac{\frac{1}{f}}{\pi}\right)}\right)}} \]

11. Applied egg-rr 95.1%

    \[\leadsto -\color{blue}{\frac{3}{\frac{\pi \cdot 0.25}{\log \left(\sqrt[3]{\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, -0.16666666666666666\right), \frac{\frac{4}{f}}{\pi}\right)}\right)}}} \]
12. Step-by-step derivation

    1. associate-/r/ 95.3%

       \[\leadsto -\color{blue}{\frac{3}{\pi \cdot 0.25} \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, -0.16666666666666666\right), \frac{\frac{4}{f}}{\pi}\right)}\right)} \]

    2. *-commutative 95.3%

       \[\leadsto -\frac{3}{\color{blue}{0.25 \cdot \pi}} \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, -0.16666666666666666\right), \frac{\frac{4}{f}}{\pi}\right)}\right) \]

    3. associate-/r* 95.3%

       \[\leadsto -\color{blue}{\frac{\frac{3}{0.25}}{\pi}} \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, -0.16666666666666666\right), \frac{\frac{4}{f}}{\pi}\right)}\right) \]

    4. metadata-eval 95.3%

       \[\leadsto -\frac{\color{blue}{12}}{\pi} \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, -0.16666666666666666\right), \frac{\frac{4}{f}}{\pi}\right)}\right) \]

    5. associate-/r* 95.3%

       \[\leadsto -\frac{12}{\pi} \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, -0.16666666666666666\right), \color{blue}{\frac{4}{f \cdot \pi}}\right)}\right) \]

13. Simplified 95.3%

    \[\leadsto -\color{blue}{\frac{12}{\pi} \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, -0.16666666666666666\right), \frac{4}{f \cdot \pi}\right)}\right)} \]

14. Final simplification 95.3%

    \[\leadsto \log \left(\sqrt[3]{\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, -0.16666666666666666\right), \frac{4}{\pi \cdot f}\right)}\right) \cdot \frac{-12}{\pi} \]
15. Add Preprocessing

Alternative 2: 96.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(\mathsf{fma}\left(f, -0.125 + \left(\pi \cdot 0.0625\right) \cdot 2, 4 \cdot \frac{\frac{1}{f}}{\pi}\right)\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (- (log (fma f (+ -0.125 (* (* PI 0.0625) 2.0)) (* 4.0 (/ (/ 1.0 f) PI)))))
  (* PI 0.25)))

C

double code(double f) {
	return -log(fma(f, (-0.125 + ((((double) M_PI) * 0.0625) * 2.0)), (4.0 * ((1.0 / f) / ((double) M_PI))))) / (((double) M_PI) * 0.25);
}

Julia

function code(f)
	return Float64(Float64(-log(fma(f, Float64(-0.125 + Float64(Float64(pi * 0.0625) * 2.0)), Float64(4.0 * Float64(Float64(1.0 / f) / pi))))) / Float64(pi * 0.25))
end

Wolfram

code[f_] := N[((-N[Log[N[(f * N[(-0.125 + N[(N[(Pi * 0.0625), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(1.0 / f), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.1%

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

3. Taylor expanded in f around 0 95.2%

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

5. Simplified 95.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), \frac{\frac{2}{f}}{\pi \cdot 0.5}\right)\right)} \]

7. Applied egg-rr 95.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\color{blue}{3} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), \frac{\frac{2}{f}}{\pi \cdot 0.5}\right)\right) \]
8. Step-by-step derivation

   1. associate-*l/ 95.3%

      \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(3 \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), \frac{\frac{2}{f}}{\pi \cdot 0.5}\right)\right)}{\frac{\pi}{4}}} \]

9. Applied egg-rr 95.3%

   \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, -0.125 + \left(\pi \cdot 0.0625\right) \cdot 2, 4 \cdot \frac{\frac{1}{f}}{\pi}\right)\right)}{\pi \cdot 0.25}} \]

10. Final simplification 95.3%

    \[\leadsto \frac{-\log \left(\mathsf{fma}\left(f, -0.125 + \left(\pi \cdot 0.0625\right) \cdot 2, 4 \cdot \frac{\frac{1}{f}}{\pi}\right)\right)}{\pi \cdot 0.25} \]
11. Add Preprocessing

Alternative 3: 96.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \log \left(f \cdot \left(\pi \cdot 0.125 - 0.125\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.1%

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

3. Taylor expanded in f around 0 95.2%

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

5. Simplified 95.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), \frac{\frac{2}{f}}{\pi \cdot 0.5}\right)\right)} \]

7. Applied egg-rr 95.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\color{blue}{3} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), \frac{\frac{2}{f}}{\pi \cdot 0.5}\right)\right) \]

8. Taylor expanded in f around 0 95.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(f \cdot \left(0.125 \cdot \pi - 0.125\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)} \]

9. Final simplification 95.1%

   \[\leadsto \log \left(f \cdot \left(\pi \cdot 0.125 - 0.125\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) \cdot \frac{-1}{\frac{\pi}{4}} \]
10. Add Preprocessing

Alternative 4: 95.9% accurate, 4.8× speedup

Math

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

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

Derivation

1. Initial program 6.1%

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

3. Taylor expanded in f around 0 95.2%

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

5. Simplified 95.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), \frac{\frac{2}{f}}{\pi \cdot 0.5}\right)\right)} \]

7. Applied egg-rr 95.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\color{blue}{3} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), \frac{\frac{2}{f}}{\pi \cdot 0.5}\right)\right) \]

8. Taylor expanded in f around 0 95.0%

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

   1. associate-*r/ 95.0%

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

   2. neg-mul-1 95.0%

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

   3. sub-neg 95.0%

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

   4. log-div 95.0%

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

   5. associate-*l/ 94.8%

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

10. Simplified 94.8%

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

11. Final simplification 94.8%

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

Alternative 5: 96.0% accurate, 4.8× speedup

Math

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

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(Float64(4.0 / 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[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.1%

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

3. Taylor expanded in f around 0 94.8%

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

   1. *-commutative 94.8%

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

   2. associate-/r* 94.8%

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

   3. distribute-rgt-out-- 94.8%

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

   4. metadata-eval 94.8%

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

5. Simplified 94.8%

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

6. Taylor expanded in f around 0 94.9%

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

   1. mul-1-neg 94.9%

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

   2. sub-neg 94.9%

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

8. Simplified 94.9%

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

   1. associate-*l/ 95.0%

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

   2. *-un-lft-identity 95.0%

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

   3. diff-log 95.0%

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

   4. div-inv 95.0%

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

   5. metadata-eval 95.0%

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

10. Applied egg-rr 95.0%

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

11. Final simplification 95.0%

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

Alternative 6: 95.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.1%

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

3. Taylor expanded in f around 0 94.8%

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

   1. *-commutative 94.8%

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

   2. associate-/r* 94.8%

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

   3. distribute-rgt-out-- 94.8%

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

   4. metadata-eval 94.8%

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

5. Simplified 94.8%

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

6. Taylor expanded in f around 0 94.9%

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

   1. mul-1-neg 94.9%

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

   2. sub-neg 94.9%

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

8. Simplified 94.9%

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

   1. add-log-exp 70.1%

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

   2. *-commutative 70.1%

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

   3. diff-log 70.1%

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

   4. exp-to-pow 70.1%

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

   5. clear-num 70.1%

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

10. Applied egg-rr 70.1%

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

    1. log-pow 94.8%

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

    2. clear-num 94.8%

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

    3. associate-*l/ 95.0%

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

    4. *-un-lft-identity 95.0%

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

    5. div-inv 95.0%

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

    6. metadata-eval 95.0%

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

    7. associate-/r* 95.0%

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

    8. div-inv 95.0%

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

    9. associate-*l/ 95.0%

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

    10. un-div-inv 95.0%

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

12. Applied egg-rr 95.0%

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

13. Final simplification 95.0%

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

Alternative 7: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log 0.6666666666666666}{\pi} \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ (log 0.6666666666666666) PI) (- 4.0)))

C

double code(double f) {
	return (log(0.6666666666666666) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return (Math.log(0.6666666666666666) / Math.PI) * -4.0;
}

Python

def code(f):
	return (math.log(0.6666666666666666) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(log(0.6666666666666666) / pi) * Float64(-4.0))
end

MATLAB

function tmp = code(f)
	tmp = (log(0.6666666666666666) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[Log[0.6666666666666666], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision]

Derivation

1. Initial program 6.1%

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

4. Applied egg-rr 1.7%

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

5. Taylor expanded in f around 0 1.7%

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

   1. *-commutative 1.7%

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

7. Simplified 1.7%

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

8. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.6666666666666666}{\pi}} \]

9. Final simplification 1.6%

   \[\leadsto \frac{\log 0.6666666666666666}{\pi} \cdot \left(-4\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(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)))))))))