VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 96.4%

Time: 21.5s

Alternatives: 3

Speedup: 4.9×

• 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: 6.6% 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: 96.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \frac{\pi}{\frac{\frac{12}{\pi}}{0.5}}, 0\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (fma
   4.0
   (/ (- (log (* (/ PI 4.0) f))) PI)
   (fma 2.0 (* (/ (pow f 2.0) PI) (/ PI (/ (/ 12.0 PI) 0.5))) 0.0))))

C

double code(double f) {
	return -fma(4.0, (-log(((((double) M_PI) / 4.0) * f)) / ((double) M_PI)), fma(2.0, ((pow(f, 2.0) / ((double) M_PI)) * (((double) M_PI) / ((12.0 / ((double) M_PI)) / 0.5))), 0.0));
}

Julia

function code(f)
	return Float64(-fma(4.0, Float64(Float64(-log(Float64(Float64(pi / 4.0) * f))) / pi), fma(2.0, Float64(Float64((f ^ 2.0) / pi) * Float64(pi / Float64(Float64(12.0 / pi) / 0.5))), 0.0)))
end

Wolfram

code[f_] := (-N[(4.0 * N[((-N[Log[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision]) / Pi), $MachinePrecision] + N[(2.0 * N[(N[(N[Power[f, 2.0], $MachinePrecision] / Pi), $MachinePrecision] * N[(Pi / N[(N[(12.0 / Pi), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 6.0%

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

3. Taylor expanded in f around 0 96.5%

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

5. Simplified 96.5%

   \[\leadsto -\color{blue}{\mathsf{fma}\left(4, \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), 0\right), 0\right)\right)} \]
6. Step-by-step derivation

   1. div-sub 96.4%

      \[\leadsto -\mathsf{fma}\left(4, \color{blue}{\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right)}{\pi} - \frac{\log f}{\pi}}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), 0\right), 0\right)\right) \]

7. Applied egg-rr 96.4%

   \[\leadsto -\mathsf{fma}\left(4, \color{blue}{\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right)}{\pi} - \frac{\log f}{\pi}}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), 0\right), 0\right)\right) \]
8. Step-by-step derivation

   1. div-sub 96.5%

      \[\leadsto -\mathsf{fma}\left(4, \color{blue}{\frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), 0\right), 0\right)\right) \]

   2. log-div 96.1%

      \[\leadsto -\mathsf{fma}\left(4, \frac{\color{blue}{\log \left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)}}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), 0\right), 0\right)\right) \]

9. Simplified 96.5%

   \[\leadsto -\mathsf{fma}\left(4, \color{blue}{\frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right), 0\right), 0\right)\right) \]
10. Step-by-step derivation

    1. fma-udef 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right)\right) + 0\right)}, 0\right)\right) \]

    2. pow-div 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{{\pi}^{\left(3 - 2\right)}} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right)\right) + 0\right), 0\right)\right) \]

    3. metadata-eval 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left({\pi}^{\color{blue}{1}} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right)\right) + 0\right), 0\right)\right) \]

    4. pow1 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{\pi} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5}\right)\right) + 0\right), 0\right)\right) \]

    5. div-inv 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \color{blue}{\left(0.0625 \cdot \pi\right) \cdot \frac{1}{0.5}}\right)\right) + 0\right), 0\right)\right) \]

    6. *-commutative 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \color{blue}{\left(\pi \cdot 0.0625\right)} \cdot \frac{1}{0.5}\right)\right) + 0\right), 0\right)\right) \]

    7. metadata-eval 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \left(\pi \cdot 0.0625\right) \cdot \color{blue}{2}\right)\right) + 0\right), 0\right)\right) \]

11. Applied egg-rr 96.5%

    \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \left(\pi \cdot 0.0625\right) \cdot 2\right)\right) + 0\right)}, 0\right)\right) \]
12. Step-by-step derivation

    1. +-rgt-identity 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \left(\pi \cdot 0.0625\right) \cdot 2\right)\right)\right)}, 0\right)\right) \]

    2. *-commutative 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\left(0.5 \cdot \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \left(\pi \cdot 0.0625\right) \cdot 2\right)\right) \cdot \pi\right)}, 0\right)\right) \]

    3. *-commutative 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \left(\color{blue}{\left(\mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \left(\pi \cdot 0.0625\right) \cdot 2\right) \cdot 0.5\right)} \cdot \pi\right), 0\right)\right) \]

    4. associate-*l* 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \left(\pi \cdot 0.0625\right) \cdot 2\right) \cdot \left(0.5 \cdot \pi\right)\right)}, 0\right)\right) \]

    5. fma-udef 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \left(\color{blue}{\left(\left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \left(\pi \cdot 0.0625\right) \cdot 2\right)} \cdot \left(0.5 \cdot \pi\right)\right), 0\right)\right) \]

    6. associate-*l* 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \left(\left(\color{blue}{\pi \cdot \left(0.020833333333333332 \cdot -2\right)} + \left(\pi \cdot 0.0625\right) \cdot 2\right) \cdot \left(0.5 \cdot \pi\right)\right), 0\right)\right) \]

    7. metadata-eval 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot \color{blue}{-0.041666666666666664} + \left(\pi \cdot 0.0625\right) \cdot 2\right) \cdot \left(0.5 \cdot \pi\right)\right), 0\right)\right) \]

    8. associate-*l* 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot -0.041666666666666664 + \color{blue}{\pi \cdot \left(0.0625 \cdot 2\right)}\right) \cdot \left(0.5 \cdot \pi\right)\right), 0\right)\right) \]

    9. metadata-eval 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot -0.041666666666666664 + \pi \cdot \color{blue}{0.125}\right) \cdot \left(0.5 \cdot \pi\right)\right), 0\right)\right) \]

    10. *-commutative 96.5%

        \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot -0.041666666666666664 + \pi \cdot 0.125\right) \cdot \color{blue}{\left(\pi \cdot 0.5\right)}\right), 0\right)\right) \]

13. Simplified 96.5%

    \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\left(\pi \cdot -0.041666666666666664 + \pi \cdot 0.125\right) \cdot \left(\pi \cdot 0.5\right)\right)}, 0\right)\right) \]
14. Step-by-step derivation

    1. *-commutative 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot -0.041666666666666664 + \pi \cdot 0.125\right)\right)}, 0\right)\right) \]

    2. flip-+ 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.5\right) \cdot \color{blue}{\frac{\left(\pi \cdot -0.041666666666666664\right) \cdot \left(\pi \cdot -0.041666666666666664\right) - \left(\pi \cdot 0.125\right) \cdot \left(\pi \cdot 0.125\right)}{\pi \cdot -0.041666666666666664 - \pi \cdot 0.125}}\right), 0\right)\right) \]

    3. associate-*r/ 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \color{blue}{\frac{\left(\pi \cdot 0.5\right) \cdot \left(\left(\pi \cdot -0.041666666666666664\right) \cdot \left(\pi \cdot -0.041666666666666664\right) - \left(\pi \cdot 0.125\right) \cdot \left(\pi \cdot 0.125\right)\right)}{\pi \cdot -0.041666666666666664 - \pi \cdot 0.125}}, 0\right)\right) \]

    4. difference-of-squares 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\left(\pi \cdot -0.041666666666666664 + \pi \cdot 0.125\right) \cdot \left(\pi \cdot -0.041666666666666664 - \pi \cdot 0.125\right)\right)}}{\pi \cdot -0.041666666666666664 - \pi \cdot 0.125}, 0\right)\right) \]

    5. distribute-lft-out 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \frac{\left(\pi \cdot 0.5\right) \cdot \left(\color{blue}{\left(\pi \cdot \left(-0.041666666666666664 + 0.125\right)\right)} \cdot \left(\pi \cdot -0.041666666666666664 - \pi \cdot 0.125\right)\right)}{\pi \cdot -0.041666666666666664 - \pi \cdot 0.125}, 0\right)\right) \]

    6. metadata-eval 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \frac{\left(\pi \cdot 0.5\right) \cdot \left(\left(\pi \cdot \color{blue}{0.08333333333333333}\right) \cdot \left(\pi \cdot -0.041666666666666664 - \pi \cdot 0.125\right)\right)}{\pi \cdot -0.041666666666666664 - \pi \cdot 0.125}, 0\right)\right) \]

    7. distribute-lft-out-- 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \frac{\left(\pi \cdot 0.5\right) \cdot \left(\left(\pi \cdot 0.08333333333333333\right) \cdot \color{blue}{\left(\pi \cdot \left(-0.041666666666666664 - 0.125\right)\right)}\right)}{\pi \cdot -0.041666666666666664 - \pi \cdot 0.125}, 0\right)\right) \]

    8. metadata-eval 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \frac{\left(\pi \cdot 0.5\right) \cdot \left(\left(\pi \cdot 0.08333333333333333\right) \cdot \left(\pi \cdot \color{blue}{-0.16666666666666666}\right)\right)}{\pi \cdot -0.041666666666666664 - \pi \cdot 0.125}, 0\right)\right) \]

    9. distribute-lft-out-- 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \frac{\left(\pi \cdot 0.5\right) \cdot \left(\left(\pi \cdot 0.08333333333333333\right) \cdot \left(\pi \cdot -0.16666666666666666\right)\right)}{\color{blue}{\pi \cdot \left(-0.041666666666666664 - 0.125\right)}}, 0\right)\right) \]

    10. metadata-eval 96.5%

        \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \frac{\left(\pi \cdot 0.5\right) \cdot \left(\left(\pi \cdot 0.08333333333333333\right) \cdot \left(\pi \cdot -0.16666666666666666\right)\right)}{\pi \cdot \color{blue}{-0.16666666666666666}}, 0\right)\right) \]

15. Applied egg-rr 96.5%

    \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \color{blue}{\frac{\left(\pi \cdot 0.5\right) \cdot \left(\left(\pi \cdot 0.08333333333333333\right) \cdot \left(\pi \cdot -0.16666666666666666\right)\right)}{\pi \cdot -0.16666666666666666}}, 0\right)\right) \]
16. Step-by-step derivation

    1. associate-/l* 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \color{blue}{\frac{\pi \cdot 0.5}{\frac{\pi \cdot -0.16666666666666666}{\left(\pi \cdot 0.08333333333333333\right) \cdot \left(\pi \cdot -0.16666666666666666\right)}}}, 0\right)\right) \]

    2. associate-/l* 96.5%

       \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \color{blue}{\frac{\pi}{\frac{\frac{\pi \cdot -0.16666666666666666}{\left(\pi \cdot 0.08333333333333333\right) \cdot \left(\pi \cdot -0.16666666666666666\right)}}{0.5}}}, 0\right)\right) \]

17. Simplified 96.5%

    \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \color{blue}{\frac{\pi}{\frac{\frac{12}{\pi}}{0.5}}}, 0\right)\right) \]

18. Final simplification 96.5%

    \[\leadsto -\mathsf{fma}\left(4, \frac{-\log \left(\frac{\pi}{4} \cdot f\right)}{\pi}, \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \frac{\pi}{\frac{\frac{12}{\pi}}{0.5}}, 0\right)\right) \]
19. Add Preprocessing

Alternative 2: 95.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.0%

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

3. Taylor expanded in f around 0 95.7%

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

   1. associate-/r* 95.3%

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

   2. distribute-rgt-out-- 95.3%

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

   3. metadata-eval 95.3%

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

5. Simplified 95.3%

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

   1. *-commutative 95.3%

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

   2. add-log-exp 72.8%

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

   3. exp-to-pow 72.8%

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

   4. *-un-lft-identity 72.8%

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

   5. *-commutative 72.8%

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

   6. times-frac 72.8%

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

   7. metadata-eval 72.8%

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

   8. inv-pow 72.8%

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

   9. div-inv 72.8%

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

   10. metadata-eval 72.8%

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

   11. *-commutative 72.8%

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

   12. unpow-prod-down 72.8%

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

   13. metadata-eval 72.8%

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

   14. inv-pow 72.8%

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

7. Applied egg-rr 72.8%

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

   1. log-pow 95.3%

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

   2. associate-*r/ 95.3%

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

   3. metadata-eval 95.3%

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

   4. associate-*r/ 95.0%

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

   5. associate-*r/ 95.0%

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

   6. metadata-eval 95.0%

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

9. Simplified 95.0%

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

    1. associate-*l/ 95.1%

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

    2. associate-/l/ 95.8%

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

11. Applied egg-rr 95.8%

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

    1. expm1-log1p-u 94.5%

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

    2. expm1-udef 94.5%

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

13. Applied egg-rr 94.9%

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

    1. expm1-def 94.9%

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

    2. expm1-log1p 96.1%

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

15. Simplified 96.1%

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

16. Final simplification 96.1%

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

Alternative 3: 95.9% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.0%

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

3. Taylor expanded in f around 0 95.7%

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

4. Taylor expanded in f around 0 96.2%

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

6. Simplified 96.2%

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

7. Final simplification 96.2%

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

Reproduce

herbie shell --seed 2024040 
(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)))))))))