VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 96.7%

Time: 28.5s

Alternatives: 9

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: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.08333333333333333\right) \cdot \left(\pi \cdot 0.5\right)\right), -2, 0\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (fma
  (/ (- (log (/ 4.0 PI)) (log f)) PI)
  -4.0
  (fma
   (* (/ (pow f 2.0) PI) (* (* PI 0.08333333333333333) (* PI 0.5)))
   -2.0
   0.0)))

C

double code(double f) {
	return fma(((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI)), -4.0, fma(((pow(f, 2.0) / ((double) M_PI)) * ((((double) M_PI) * 0.08333333333333333) * (((double) M_PI) * 0.5))), -2.0, 0.0));
}

Julia

function code(f)
	return fma(Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi), -4.0, fma(Float64(Float64((f ^ 2.0) / pi) * Float64(Float64(pi * 0.08333333333333333) * Float64(pi * 0.5))), -2.0, 0.0))
end

Wolfram

code[f_] := N[(N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * -4.0 + N[(N[(N[(N[Power[f, 2.0], $MachinePrecision] / Pi), $MachinePrecision] * N[(N[(Pi * 0.08333333333333333), $MachinePrecision] * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

   1. distribute-lft-neg-in 7.9%

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

   2. *-commutative 7.9%

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

3. Simplified 7.9%

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

5. Taylor expanded in f around 0 97.0%

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

7. Simplified 97.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \left(2 \cdot \pi\right)\right), 0\right), -2, 0\right)\right)} \]
8. Step-by-step derivation

   1. fma-udef 97.0%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \left(2 \cdot \pi\right)\right)\right) + 0\right)}, -2, 0\right)\right) \]

   2. associate-/r* 97.0%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{0.005208333333333333}{0.5}}{\frac{0.5}{\pi}}}, -2, 0.0625 \cdot \left(2 \cdot \pi\right)\right)\right) + 0\right), -2, 0\right)\right) \]

   3. metadata-eval 97.0%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\frac{\color{blue}{0.010416666666666666}}{\frac{0.5}{\pi}}, -2, 0.0625 \cdot \left(2 \cdot \pi\right)\right)\right) + 0\right), -2, 0\right)\right) \]

   4. associate-*r* 97.0%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, \color{blue}{\left(0.0625 \cdot 2\right) \cdot \pi}\right)\right) + 0\right), -2, 0\right)\right) \]

   5. metadata-eval 97.0%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, \color{blue}{0.125} \cdot \pi\right)\right) + 0\right), -2, 0\right)\right) \]

9. Applied egg-rr 97.0%

   \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, 0.125 \cdot \pi\right)\right) + 0\right)}, -2, 0\right)\right) \]
10. Step-by-step derivation

    1. +-rgt-identity 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, 0.125 \cdot \pi\right)\right)\right)}, -2, 0\right)\right) \]

    2. associate-*r* 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, 0.125 \cdot \pi\right)\right)}, -2, 0\right)\right) \]

    3. fma-def 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}} \cdot -2 + 0.125 \cdot \pi\right)}\right), -2, 0\right)\right) \]

    4. +-commutative 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(0.125 \cdot \pi + \frac{0.010416666666666666}{\frac{0.5}{\pi}} \cdot -2\right)}\right), -2, 0\right)\right) \]

    5. *-commutative 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.5\right) \cdot \left(\color{blue}{\pi \cdot 0.125} + \frac{0.010416666666666666}{\frac{0.5}{\pi}} \cdot -2\right)\right), -2, 0\right)\right) \]

    6. fma-def 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(\pi, 0.125, \frac{0.010416666666666666}{\frac{0.5}{\pi}} \cdot -2\right)}\right), -2, 0\right)\right) \]

    7. *-commutative 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, \color{blue}{-2 \cdot \frac{0.010416666666666666}{\frac{0.5}{\pi}}}\right)\right), -2, 0\right)\right) \]

    8. associate-/r/ 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, -2 \cdot \color{blue}{\left(\frac{0.010416666666666666}{0.5} \cdot \pi\right)}\right)\right), -2, 0\right)\right) \]

    9. associate-*r* 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, \color{blue}{\left(-2 \cdot \frac{0.010416666666666666}{0.5}\right) \cdot \pi}\right)\right), -2, 0\right)\right) \]

    10. metadata-eval 97.0%

        \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, \left(-2 \cdot \color{blue}{0.020833333333333332}\right) \cdot \pi\right)\right), -2, 0\right)\right) \]

    11. metadata-eval 97.0%

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

11. Simplified 97.0%

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

    1. expm1-log1p-u 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, -0.041666666666666664 \cdot \pi\right)\right)\right)}, -2, 0\right)\right) \]

    2. expm1-udef 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, -0.041666666666666664 \cdot \pi\right)\right)} - 1\right)}, -2, 0\right)\right) \]

    3. associate-*l* 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi, 0.125, -0.041666666666666664 \cdot \pi\right)\right)}\right)} - 1\right), -2, 0\right)\right) \]

    4. *-commutative 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(e^{\mathsf{log1p}\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi, 0.125, \color{blue}{\pi \cdot -0.041666666666666664}\right)\right)\right)} - 1\right), -2, 0\right)\right) \]

13. Applied egg-rr 97.0%

    \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi, 0.125, \pi \cdot -0.041666666666666664\right)\right)\right)} - 1\right)}, -2, 0\right)\right) \]
14. Step-by-step derivation

    1. expm1-def 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi, 0.125, \pi \cdot -0.041666666666666664\right)\right)\right)\right)}, -2, 0\right)\right) \]

    2. expm1-log1p 97.0%

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

    3. *-commutative 97.0%

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

    4. *-commutative 97.0%

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

    5. associate-*l* 97.0%

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

    6. fma-udef 97.0%

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

    7. distribute-lft-out 97.0%

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

    8. metadata-eval 97.0%

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

    9. *-commutative 97.0%

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

15. Simplified 97.0%

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

16. Final simplification 97.0%

    \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.08333333333333333\right) \cdot \left(\pi \cdot 0.5\right)\right), -2, 0\right)\right) \]
17. Add Preprocessing

Alternative 2: 96.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.08333333333333333\right) \cdot \left(\pi \cdot 0.5\right)\right), -2, 0\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (fma
  (/ (log (/ 4.0 (* PI f))) PI)
  -4.0
  (fma
   (* (/ (pow f 2.0) PI) (* (* PI 0.08333333333333333) (* PI 0.5)))
   -2.0
   0.0)))

C

double code(double f) {
	return fma((log((4.0 / (((double) M_PI) * f))) / ((double) M_PI)), -4.0, fma(((pow(f, 2.0) / ((double) M_PI)) * ((((double) M_PI) * 0.08333333333333333) * (((double) M_PI) * 0.5))), -2.0, 0.0));
}

Julia

function code(f)
	return fma(Float64(log(Float64(4.0 / Float64(pi * f))) / pi), -4.0, fma(Float64(Float64((f ^ 2.0) / pi) * Float64(Float64(pi * 0.08333333333333333) * Float64(pi * 0.5))), -2.0, 0.0))
end

Wolfram

code[f_] := N[(N[(N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * -4.0 + N[(N[(N[(N[Power[f, 2.0], $MachinePrecision] / Pi), $MachinePrecision] * N[(N[(Pi * 0.08333333333333333), $MachinePrecision] * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

   1. distribute-lft-neg-in 7.9%

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

   2. *-commutative 7.9%

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

3. Simplified 7.9%

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

5. Taylor expanded in f around 0 97.0%

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

7. Simplified 97.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \left(2 \cdot \pi\right)\right), 0\right), -2, 0\right)\right)} \]
8. Step-by-step derivation

   1. fma-udef 97.0%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \left(2 \cdot \pi\right)\right)\right) + 0\right)}, -2, 0\right)\right) \]

   2. associate-/r* 97.0%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{0.005208333333333333}{0.5}}{\frac{0.5}{\pi}}}, -2, 0.0625 \cdot \left(2 \cdot \pi\right)\right)\right) + 0\right), -2, 0\right)\right) \]

   3. metadata-eval 97.0%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\frac{\color{blue}{0.010416666666666666}}{\frac{0.5}{\pi}}, -2, 0.0625 \cdot \left(2 \cdot \pi\right)\right)\right) + 0\right), -2, 0\right)\right) \]

   4. associate-*r* 97.0%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, \color{blue}{\left(0.0625 \cdot 2\right) \cdot \pi}\right)\right) + 0\right), -2, 0\right)\right) \]

   5. metadata-eval 97.0%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, \color{blue}{0.125} \cdot \pi\right)\right) + 0\right), -2, 0\right)\right) \]

9. Applied egg-rr 97.0%

   \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, 0.125 \cdot \pi\right)\right) + 0\right)}, -2, 0\right)\right) \]
10. Step-by-step derivation

    1. +-rgt-identity 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, 0.125 \cdot \pi\right)\right)\right)}, -2, 0\right)\right) \]

    2. associate-*r* 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}}, -2, 0.125 \cdot \pi\right)\right)}, -2, 0\right)\right) \]

    3. fma-def 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\frac{0.010416666666666666}{\frac{0.5}{\pi}} \cdot -2 + 0.125 \cdot \pi\right)}\right), -2, 0\right)\right) \]

    4. +-commutative 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(0.125 \cdot \pi + \frac{0.010416666666666666}{\frac{0.5}{\pi}} \cdot -2\right)}\right), -2, 0\right)\right) \]

    5. *-commutative 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.5\right) \cdot \left(\color{blue}{\pi \cdot 0.125} + \frac{0.010416666666666666}{\frac{0.5}{\pi}} \cdot -2\right)\right), -2, 0\right)\right) \]

    6. fma-def 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(\pi, 0.125, \frac{0.010416666666666666}{\frac{0.5}{\pi}} \cdot -2\right)}\right), -2, 0\right)\right) \]

    7. *-commutative 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, \color{blue}{-2 \cdot \frac{0.010416666666666666}{\frac{0.5}{\pi}}}\right)\right), -2, 0\right)\right) \]

    8. associate-/r/ 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, -2 \cdot \color{blue}{\left(\frac{0.010416666666666666}{0.5} \cdot \pi\right)}\right)\right), -2, 0\right)\right) \]

    9. associate-*r* 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, \color{blue}{\left(-2 \cdot \frac{0.010416666666666666}{0.5}\right) \cdot \pi}\right)\right), -2, 0\right)\right) \]

    10. metadata-eval 97.0%

        \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, \left(-2 \cdot \color{blue}{0.020833333333333332}\right) \cdot \pi\right)\right), -2, 0\right)\right) \]

    11. metadata-eval 97.0%

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

11. Simplified 97.0%

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

    1. expm1-log1p-u 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, -0.041666666666666664 \cdot \pi\right)\right)\right)}, -2, 0\right)\right) \]

    2. expm1-udef 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, -0.041666666666666664 \cdot \pi\right)\right)} - 1\right)}, -2, 0\right)\right) \]

    3. associate-*l* 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi, 0.125, -0.041666666666666664 \cdot \pi\right)\right)}\right)} - 1\right), -2, 0\right)\right) \]

    4. *-commutative 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(e^{\mathsf{log1p}\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi, 0.125, \color{blue}{\pi \cdot -0.041666666666666664}\right)\right)\right)} - 1\right), -2, 0\right)\right) \]

13. Applied egg-rr 97.0%

    \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi, 0.125, \pi \cdot -0.041666666666666664\right)\right)\right)} - 1\right)}, -2, 0\right)\right) \]
14. Step-by-step derivation

    1. expm1-def 97.0%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi, 0.125, \pi \cdot -0.041666666666666664\right)\right)\right)\right)}, -2, 0\right)\right) \]

    2. expm1-log1p 97.0%

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

    3. *-commutative 97.0%

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

    4. *-commutative 97.0%

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

    5. associate-*l* 97.0%

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

    6. fma-udef 97.0%

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

    7. distribute-lft-out 97.0%

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

    8. metadata-eval 97.0%

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

    9. *-commutative 97.0%

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

15. Simplified 97.0%

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

16. Taylor expanded in f around inf 97.0%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(\frac{4}{\pi}\right) - -1 \cdot \log \left(\frac{1}{f}\right)}{\pi}}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.08333333333333333\right) \cdot \left(\pi \cdot 0.5\right)\right), -2, 0\right)\right) \]
17. Step-by-step derivation

    1. div-sub 96.9%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\log \left(\frac{4}{\pi}\right)}{\pi} - \frac{-1 \cdot \log \left(\frac{1}{f}\right)}{\pi}}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.08333333333333333\right) \cdot \left(\pi \cdot 0.5\right)\right), -2, 0\right)\right) \]

    2. mul-1-neg 96.9%

       \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right)}{\pi} - \frac{\color{blue}{-\log \left(\frac{1}{f}\right)}}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.08333333333333333\right) \cdot \left(\pi \cdot 0.5\right)\right), -2, 0\right)\right) \]

    3. log-rec 96.9%

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

    4. remove-double-neg 96.9%

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

    5. div-sub 97.0%

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

    6. log-div 97.0%

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

    7. associate-/r* 97.0%

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

18. Simplified 97.0%

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

19. Final simplification 97.0%

    \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.08333333333333333\right) \cdot \left(\pi \cdot 0.5\right)\right), -2, 0\right)\right) \]
20. Add Preprocessing

Alternative 3: 96.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \left(\pi \cdot 2\right)\right), \frac{\frac{4}{\pi}}{f}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (fma
    f
    (fma
     (/ 0.005208333333333333 (* 0.5 (/ 0.5 PI)))
     -2.0
     (* 0.0625 (* PI 2.0)))
    (/ (/ 4.0 PI) f)))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(fma(f, fma((0.005208333333333333 / (0.5 * (0.5 / ((double) M_PI)))), -2.0, (0.0625 * (((double) M_PI) * 2.0))), ((4.0 / ((double) M_PI)) / f))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(fma(f, fma(Float64(0.005208333333333333 / Float64(0.5 * Float64(0.5 / pi))), -2.0, Float64(0.0625 * Float64(pi * 2.0))), Float64(Float64(4.0 / pi) / f))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(N[(0.005208333333333333 / N[(0.5 * N[(0.5 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + N[(0.0625 * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

3. Taylor expanded in f around 0 96.9%

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

5. Simplified 96.9%

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

6. Final simplification 96.9%

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

Alternative 4: 96.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\log \left(\frac{4}{\pi \cdot f}\right) + \mathsf{fma}\left({f}^{2} \cdot 0.5, \pi \cdot \left(\pi \cdot 0.041666666666666664\right), f \cdot 0\right)\right) \cdot \frac{-4}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (+
   (log (/ 4.0 (* PI f)))
   (fma (* (pow f 2.0) 0.5) (* PI (* PI 0.041666666666666664)) (* f 0.0)))
  (/ -4.0 PI)))

C

double code(double f) {
	return (log((4.0 / (((double) M_PI) * f))) + fma((pow(f, 2.0) * 0.5), (((double) M_PI) * (((double) M_PI) * 0.041666666666666664)), (f * 0.0))) * (-4.0 / ((double) M_PI));
}

Julia

function code(f)
	return Float64(Float64(log(Float64(4.0 / Float64(pi * f))) + fma(Float64((f ^ 2.0) * 0.5), Float64(pi * Float64(pi * 0.041666666666666664)), Float64(f * 0.0))) * Float64(-4.0 / pi))
end

Wolfram

code[f_] := N[(N[(N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Power[f, 2.0], $MachinePrecision] * 0.5), $MachinePrecision] * N[(Pi * N[(Pi * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + N[(f * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

   1. distribute-lft-neg-in 7.9%

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

   2. *-commutative 7.9%

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

3. Simplified 7.9%

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

5. Taylor expanded in f around inf 7.9%

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

6. Taylor expanded in f around 0 96.9%

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

   1. fma-def 96.9%

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

   2. distribute-rgt-out-- 96.9%

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

   3. metadata-eval 96.9%

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

   4. distribute-rgt-out-- 96.9%

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

   5. metadata-eval 96.9%

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

   6. associate-*r* 96.9%

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

   7. *-commutative 96.9%

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

   8. cube-prod 96.9%

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

8. Simplified 96.9%

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

9. Taylor expanded in f around 0 97.0%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]

11. Simplified 96.9%

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

12. Final simplification 96.9%

    \[\leadsto \left(\log \left(\frac{4}{\pi \cdot f}\right) + \mathsf{fma}\left({f}^{2} \cdot 0.5, \pi \cdot \left(\pi \cdot 0.041666666666666664\right), f \cdot 0\right)\right) \cdot \frac{-4}{\pi} \]
13. Add Preprocessing

Alternative 5: 96.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{\log f - \log \left(\frac{4}{\pi}\right)}{\pi} - 0.125 \cdot \left(\pi \cdot {f}^{2}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (- (* 4.0 (/ (- (log f) (log (/ 4.0 PI))) PI)) (* 0.125 (* PI (pow f 2.0)))))

C

double code(double f) {
	return (4.0 * ((log(f) - log((4.0 / ((double) M_PI)))) / ((double) M_PI))) - (0.125 * (((double) M_PI) * pow(f, 2.0)));
}

Java

public static double code(double f) {
	return (4.0 * ((Math.log(f) - Math.log((4.0 / Math.PI))) / Math.PI)) - (0.125 * (Math.PI * Math.pow(f, 2.0)));
}

Python

def code(f):
	return (4.0 * ((math.log(f) - math.log((4.0 / math.pi))) / math.pi)) - (0.125 * (math.pi * math.pow(f, 2.0)))

Julia

function code(f)
	return Float64(Float64(4.0 * Float64(Float64(log(f) - log(Float64(4.0 / pi))) / pi)) - Float64(0.125 * Float64(pi * (f ^ 2.0))))
end

MATLAB

function tmp = code(f)
	tmp = (4.0 * ((log(f) - log((4.0 / pi))) / pi)) - (0.125 * (pi * (f ^ 2.0)));
end

Wolfram

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

Derivation

1. Initial program 7.9%

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

3. Taylor expanded in f around 0 96.3%

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

   1. distribute-rgt-out-- 96.3%

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

   2. metadata-eval 96.3%

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

5. Simplified 96.3%

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

6. Taylor expanded in f around 0 96.4%

   \[\leadsto -\color{blue}{\left(0.125 \cdot \left({f}^{2} \cdot \pi\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}{\pi}\right)} \]

7. Final simplification 96.4%

   \[\leadsto 4 \cdot \frac{\log f - \log \left(\frac{4}{\pi}\right)}{\pi} - 0.125 \cdot \left(\pi \cdot {f}^{2}\right) \]
8. Add Preprocessing

Alternative 6: 96.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(4 \cdot \left(0.03125 \cdot \left(\pi \cdot f\right) + \frac{1}{\pi \cdot f}\right)\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (- (log (* 4.0 (+ (* 0.03125 (* PI f)) (/ 1.0 (* PI f)))))) (* PI 0.25)))

C

double code(double f) {
	return -log((4.0 * ((0.03125 * (((double) M_PI) * f)) + (1.0 / (((double) M_PI) * f))))) / (((double) M_PI) * 0.25);
}

Java

public static double code(double f) {
	return -Math.log((4.0 * ((0.03125 * (Math.PI * f)) + (1.0 / (Math.PI * f))))) / (Math.PI * 0.25);
}

Python

def code(f):
	return -math.log((4.0 * ((0.03125 * (math.pi * f)) + (1.0 / (math.pi * f))))) / (math.pi * 0.25)

Julia

function code(f)
	return Float64(Float64(-log(Float64(4.0 * Float64(Float64(0.03125 * Float64(pi * f)) + Float64(1.0 / Float64(pi * f)))))) / Float64(pi * 0.25))
end

MATLAB

function tmp = code(f)
	tmp = -log((4.0 * ((0.03125 * (pi * f)) + (1.0 / (pi * f))))) / (pi * 0.25);
end

Wolfram

code[f_] := N[((-N[Log[N[(4.0 * N[(N[(0.03125 * N[(Pi * f), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

3. Taylor expanded in f around 0 96.3%

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

   1. distribute-rgt-out-- 96.3%

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

   2. metadata-eval 96.3%

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

5. Simplified 96.3%

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

   1. associate-*l/ 96.4%

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

   2. *-un-lft-identity 96.4%

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

   3. cosh-undef 96.4%

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

   4. associate-*l/ 96.4%

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

   5. div-inv 96.4%

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

   6. metadata-eval 96.4%

      \[\leadsto -\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{f \cdot \left(\pi \cdot 0.5\right)}\right)}{\pi \cdot \color{blue}{0.25}} \]

7. Applied egg-rr 96.4%

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

   1. associate-*r* 96.4%

      \[\leadsto -\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{\color{blue}{\left(f \cdot \pi\right) \cdot 0.5}}\right)}{\pi \cdot 0.25} \]

   2. *-commutative 96.4%

      \[\leadsto -\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{\color{blue}{0.5 \cdot \left(f \cdot \pi\right)}}\right)}{\pi \cdot 0.25} \]

   3. *-commutative 96.4%

      \[\leadsto -\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{0.5 \cdot \color{blue}{\left(\pi \cdot f\right)}}\right)}{\pi \cdot 0.25} \]

   4. times-frac 96.4%

      \[\leadsto -\frac{\log \color{blue}{\left(\frac{2}{0.5} \cdot \frac{\cosh \left(\frac{\pi \cdot f}{4}\right)}{\pi \cdot f}\right)}}{\pi \cdot 0.25} \]

   5. metadata-eval 96.4%

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

9. Simplified 96.4%

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

10. Taylor expanded in f around 0 96.4%

    \[\leadsto -\frac{\log \left(4 \cdot \color{blue}{\left(0.03125 \cdot \left(f \cdot \pi\right) + \frac{1}{f \cdot \pi}\right)}\right)}{\pi \cdot 0.25} \]

11. Final simplification 96.4%

    \[\leadsto \frac{-\log \left(4 \cdot \left(0.03125 \cdot \left(\pi \cdot f\right) + \frac{1}{\pi \cdot f}\right)\right)}{\pi \cdot 0.25} \]
12. Add Preprocessing

Alternative 7: 96.1% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. distribute-lft-neg-in 7.9%

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

   2. *-commutative 7.9%

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

3. Simplified 7.9%

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

5. Taylor expanded in f around 0 96.4%

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

   1. *-commutative 96.4%

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

   2. associate-*l/ 96.4%

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

   3. mul-1-neg 96.4%

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

   4. unsub-neg 96.4%

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

   5. distribute-rgt-out-- 96.4%

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

   6. metadata-eval 96.4%

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

7. Simplified 96.4%

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

8. Taylor expanded in f around 0 96.4%

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

   1. log-div 96.4%

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

   2. associate-/r* 96.4%

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

10. Simplified 96.4%

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

    1. frac-2neg 96.4%

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

    2. div-inv 96.2%

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

    3. neg-log 96.2%

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

    4. clear-num 96.2%

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

    5. div-inv 96.2%

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

    6. metadata-eval 96.2%

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

12. Applied egg-rr 96.2%

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

    1. associate-*r/ 96.4%

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

    2. *-rgt-identity 96.4%

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

    3. associate-*l* 96.4%

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

14. Simplified 96.4%

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

15. Final simplification 96.4%

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

Alternative 8: 96.1% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. distribute-lft-neg-in 7.9%

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

   2. *-commutative 7.9%

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

3. Simplified 7.9%

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

5. Taylor expanded in f around 0 96.4%

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

   1. *-commutative 96.4%

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

   2. associate-*l/ 96.4%

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

   3. mul-1-neg 96.4%

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

   4. unsub-neg 96.4%

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

   5. distribute-rgt-out-- 96.4%

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

   6. metadata-eval 96.4%

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

7. Simplified 96.4%

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

8. Taylor expanded in f around 0 96.4%

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

   1. log-div 96.4%

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

   2. associate-/r* 96.4%

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

10. Simplified 96.4%

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

11. Final simplification 96.4%

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

Alternative 9: 4.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \left({f}^{2} \cdot \left(-0.125\right)\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* PI (* (pow f 2.0) (- 0.125))))

C

double code(double f) {
	return ((double) M_PI) * (pow(f, 2.0) * -0.125);
}

Java

public static double code(double f) {
	return Math.PI * (Math.pow(f, 2.0) * -0.125);
}

Python

def code(f):
	return math.pi * (math.pow(f, 2.0) * -0.125)

Julia

function code(f)
	return Float64(pi * Float64((f ^ 2.0) * Float64(-0.125)))
end

MATLAB

function tmp = code(f)
	tmp = pi * ((f ^ 2.0) * -0.125);
end

Wolfram

code[f_] := N[(Pi * N[(N[Power[f, 2.0], $MachinePrecision] * (-0.125)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

3. Taylor expanded in f around 0 96.3%

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

   1. distribute-rgt-out-- 96.3%

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

   2. metadata-eval 96.3%

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

5. Simplified 96.3%

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

6. Taylor expanded in f around 0 96.4%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + 0.03125 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative 96.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\left(-1 \cdot \log f + 0.03125 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) + \log \left(\frac{4}{\pi}\right)\right)} \]

   2. neg-mul-1 96.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\left(\color{blue}{\left(-\log f\right)} + 0.03125 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) + \log \left(\frac{4}{\pi}\right)\right) \]

   3. +-commutative 96.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\color{blue}{\left(0.03125 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \left(-\log f\right)\right)} + \log \left(\frac{4}{\pi}\right)\right) \]

   4. associate-+l+ 96.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(0.03125 \cdot \left({f}^{2} \cdot {\pi}^{2}\right) + \left(\left(-\log f\right) + \log \left(\frac{4}{\pi}\right)\right)\right)} \]

   5. *-commutative 96.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0.03125 \cdot \color{blue}{\left({\pi}^{2} \cdot {f}^{2}\right)} + \left(\left(-\log f\right) + \log \left(\frac{4}{\pi}\right)\right)\right) \]

   6. unpow2 96.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0.03125 \cdot \left(\color{blue}{\left(\pi \cdot \pi\right)} \cdot {f}^{2}\right) + \left(\left(-\log f\right) + \log \left(\frac{4}{\pi}\right)\right)\right) \]

   7. unpow2 96.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0.03125 \cdot \left(\left(\pi \cdot \pi\right) \cdot \color{blue}{\left(f \cdot f\right)}\right) + \left(\left(-\log f\right) + \log \left(\frac{4}{\pi}\right)\right)\right) \]

   8. swap-sqr 96.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0.03125 \cdot \color{blue}{\left(\left(\pi \cdot f\right) \cdot \left(\pi \cdot f\right)\right)} + \left(\left(-\log f\right) + \log \left(\frac{4}{\pi}\right)\right)\right) \]

   9. unpow1 96.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0.03125 \cdot \left(\color{blue}{{\left(\pi \cdot f\right)}^{1}} \cdot \left(\pi \cdot f\right)\right) + \left(\left(-\log f\right) + \log \left(\frac{4}{\pi}\right)\right)\right) \]

   10. pow-plus 96.4%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0.03125 \cdot \color{blue}{{\left(\pi \cdot f\right)}^{\left(1 + 1\right)}} + \left(\left(-\log f\right) + \log \left(\frac{4}{\pi}\right)\right)\right) \]

   11. metadata-eval 96.4%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0.03125 \cdot {\left(\pi \cdot f\right)}^{\color{blue}{2}} + \left(\left(-\log f\right) + \log \left(\frac{4}{\pi}\right)\right)\right) \]

   12. +-commutative 96.4%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0.03125 \cdot {\left(\pi \cdot f\right)}^{2} + \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-\log f\right)\right)}\right) \]

   13. sub-neg 96.4%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0.03125 \cdot {\left(\pi \cdot f\right)}^{2} + \color{blue}{\left(\log \left(\frac{4}{\pi}\right) - \log f\right)}\right) \]

   14. log-div 96.3%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0.03125 \cdot {\left(\pi \cdot f\right)}^{2} + \color{blue}{\log \left(\frac{\frac{4}{\pi}}{f}\right)}\right) \]

   15. associate-/r* 96.3%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0.03125 \cdot {\left(\pi \cdot f\right)}^{2} + \log \color{blue}{\left(\frac{4}{\pi \cdot f}\right)}\right) \]

8. Simplified 96.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(0.03125 \cdot {\left(\pi \cdot f\right)}^{2} + \log \left(\frac{4}{\pi \cdot f}\right)\right)} \]

9. Taylor expanded in f around inf 4.3%

   \[\leadsto -\color{blue}{0.125 \cdot \left({f}^{2} \cdot \pi\right)} \]
10. Step-by-step derivation

    1. associate-*r* 4.3%

       \[\leadsto -\color{blue}{\left(0.125 \cdot {f}^{2}\right) \cdot \pi} \]

11. Simplified 4.3%

    \[\leadsto -\color{blue}{\left(0.125 \cdot {f}^{2}\right) \cdot \pi} \]

12. Final simplification 4.3%

    \[\leadsto \pi \cdot \left({f}^{2} \cdot \left(-0.125\right)\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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)))))))))