VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 96.6%

Time: 35.0s

Alternatives: 6

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: 6.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t\_0}\\ t_2 := e^{-t\_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t\_1 + t\_2}{t\_1 - t\_2}\right) \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \left(\left(\pi \cdot 0.5\right) \cdot \mathsf{fma}\left(\pi, 0.125, \pi \cdot -0.041666666666666664\right)\right), -2, 0\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (fma
  (/ (- (log (/ 4.0 PI)) (log f)) PI)
  -4.0
  (fma
   (*
    (/ (pow f 2.0) PI)
    (* (* PI 0.5) (fma PI 0.125 (* PI -0.041666666666666664))))
   -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) * 0.5) * fma(((double) M_PI), 0.125, (((double) M_PI) * -0.041666666666666664)))), -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 * 0.5) * fma(pi, 0.125, Float64(pi * -0.041666666666666664)))), -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 * 0.5), $MachinePrecision] * N[(Pi * 0.125 + N[(Pi * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.4%

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

   1. distribute-lft-neg-in 7.4%

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

   2. *-commutative 7.4%

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

3. Simplified 7.0%

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

5. Taylor expanded in f around 0 97.3%

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

7. Simplified 97.3%

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

   1. fma-udef 97.3%

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

   2. associate-/r* 97.3%

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

   3. metadata-eval 97.3%

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

   4. associate-*r* 97.3%

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

   5. metadata-eval 97.3%

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

9. Applied egg-rr 97.3%

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

    1. +-rgt-identity 97.3%

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

    2. associate-*r* 97.3%

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

    3. fma-def 97.3%

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

    4. +-commutative 97.3%

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

    5. *-commutative 97.3%

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

    6. fma-def 97.3%

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

    7. *-commutative 97.3%

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

    8. associate-/r/ 97.3%

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

    9. associate-*r* 97.3%

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

    10. metadata-eval 97.3%

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

    11. metadata-eval 97.3%

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

11. Simplified 97.3%

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

12. Final simplification 97.3%

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

Alternative 2: 96.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(f, -2 \cdot \frac{0.010416666666666666}{\frac{0.5}{\pi}} + \pi \cdot 0.125, \frac{\frac{4}{\pi}}{f}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (fma
    f
    (+ (* -2.0 (/ 0.010416666666666666 (/ 0.5 PI))) (* PI 0.125))
    (/ (/ 4.0 PI) f)))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(fma(f, ((-2.0 * (0.010416666666666666 / (0.5 / ((double) M_PI)))) + (((double) M_PI) * 0.125)), ((4.0 / ((double) M_PI)) / f))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(fma(f, Float64(Float64(-2.0 * Float64(0.010416666666666666 / Float64(0.5 / pi))) + Float64(pi * 0.125)), Float64(Float64(4.0 / pi) / f))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(N[(-2.0 * N[(0.010416666666666666 / N[(0.5 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.125), $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 7.4%

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

3. Taylor expanded in f around 0 97.2%

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

5. Simplified 97.2%

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

   1. fma-udef 97.2%

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

   2. associate-/r* 97.2%

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

   3. metadata-eval 97.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \frac{\color{blue}{0.010416666666666666}}{\frac{0.5}{\pi}} \cdot -2 + 0.0625 \cdot \left(2 \cdot \pi\right), \frac{\frac{4}{\pi}}{f}\right)\right) \]

   4. associate-*r* 97.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \frac{0.010416666666666666}{\frac{0.5}{\pi}} \cdot -2 + \color{blue}{\left(0.0625 \cdot 2\right) \cdot \pi}, \frac{\frac{4}{\pi}}{f}\right)\right) \]

   5. metadata-eval 97.2%

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

7. Applied egg-rr 97.2%

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

8. Final simplification 97.2%

   \[\leadsto \log \left(\mathsf{fma}\left(f, -2 \cdot \frac{0.010416666666666666}{\frac{0.5}{\pi}} + \pi \cdot 0.125, \frac{\frac{4}{\pi}}{f}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \]
9. Add Preprocessing

Alternative 3: 96.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

3. Taylor expanded in f around 0 96.8%

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

   1. distribute-rgt-out-- 96.8%

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

   2. metadata-eval 96.8%

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

5. Simplified 96.8%

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

   1. associate-*l/ 96.9%

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

   2. *-un-lft-identity 96.9%

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

   3. cosh-undef 96.9%

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

   4. *-commutative 96.9%

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

   5. div-inv 96.9%

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

   6. metadata-eval 96.9%

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

   7. div-inv 96.9%

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

   8. metadata-eval 96.9%

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

7. Applied egg-rr 96.9%

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

   1. *-lft-identity 96.9%

      \[\leadsto -\frac{\color{blue}{1 \cdot \log \left(\frac{2 \cdot \cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{f \cdot \left(\pi \cdot 0.5\right)}\right)}}{\pi \cdot 0.25} \]

   2. *-commutative 96.9%

      \[\leadsto -\frac{1 \cdot \log \left(\frac{2 \cdot \cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{f \cdot \left(\pi \cdot 0.5\right)}\right)}{\color{blue}{0.25 \cdot \pi}} \]

   3. times-frac 96.9%

      \[\leadsto -\color{blue}{\frac{1}{0.25} \cdot \frac{\log \left(\frac{2 \cdot \cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{f \cdot \left(\pi \cdot 0.5\right)}\right)}{\pi}} \]

   4. metadata-eval 96.9%

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

   5. associate-*r* 96.9%

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

   6. *-commutative 96.9%

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

   7. times-frac 96.9%

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

   8. metadata-eval 96.9%

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

   9. associate-*r* 96.9%

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

   10. *-commutative 96.9%

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

   11. associate-*r* 96.9%

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

9. Simplified 96.9%

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

10. Final simplification 96.9%

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

Alternative 4: 96.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

   1. distribute-lft-neg-in 7.4%

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

   2. *-commutative 7.4%

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

3. Simplified 7.0%

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

5. Taylor expanded in f around 0 96.9%

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

   1. *-commutative 96.9%

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

   2. associate-*l/ 96.9%

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

   3. mul-1-neg 96.9%

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

   4. unsub-neg 96.9%

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

   5. distribute-rgt-out-- 96.9%

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

   6. metadata-eval 96.9%

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

7. Simplified 96.9%

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

8. Taylor expanded in f around 0 96.9%

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

   1. div-sub 96.7%

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

   2. remove-double-neg 96.7%

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

   3. mul-1-neg 96.7%

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

   4. log-rec 96.7%

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

   5. div-sub 96.9%

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

   6. div-sub 96.7%

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

   7. log-rec 96.7%

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

   8. mul-1-neg 96.7%

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

   9. remove-double-neg 96.7%

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

   10. div-sub 96.9%

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

10. Simplified 96.9%

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

    1. log-div 96.8%

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

    2. div-sub 96.9%

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

12. Applied egg-rr 96.9%

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

13. Final simplification 96.9%

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

Alternative 5: 96.0% 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 7.4%

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

   1. distribute-lft-neg-in 7.4%

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

   2. *-commutative 7.4%

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

3. Simplified 7.0%

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

5. Taylor expanded in f around 0 96.9%

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

   1. *-commutative 96.9%

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

   2. associate-*l/ 96.9%

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

   3. mul-1-neg 96.9%

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

   4. unsub-neg 96.9%

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

   5. distribute-rgt-out-- 96.9%

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

   6. metadata-eval 96.9%

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

7. Simplified 96.9%

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

8. Taylor expanded in f around 0 96.9%

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

   1. div-sub 96.7%

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

   2. remove-double-neg 96.7%

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

   3. mul-1-neg 96.7%

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

   4. log-rec 96.7%

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

   5. div-sub 96.9%

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

   6. div-sub 96.7%

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

   7. log-rec 96.7%

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

   8. mul-1-neg 96.7%

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

   9. remove-double-neg 96.7%

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

   10. div-sub 96.9%

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

10. Simplified 96.9%

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

11. Taylor expanded in f around 0 96.9%

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

12. Final simplification 96.9%

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

Alternative 6: 0.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log 0}{\pi} \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ (log 0.0) PI) (- 4.0)))

C

double code(double f) {
	return (log(0.0) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return (Math.log(0.0) / Math.PI) * -4.0;
}

Python

def code(f):
	return (math.log(0.0) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(log(0.0) / pi) * Float64(-4.0))
end

MATLAB

function tmp = code(f)
	tmp = (log(0.0) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[Log[0.0], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision]

Derivation

1. Initial program 7.4%

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

3. Taylor expanded in f around 0 96.7%

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

4. Taylor expanded in f around inf 0.7%

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

   1. distribute-rgt-out 0.7%

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

   2. distribute-rgt-out-- 0.7%

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

   3. metadata-eval 0.7%

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

   4. metadata-eval 0.7%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{\pi}{\pi \cdot 0.5} \cdot \color{blue}{0}\right)}{\pi} \]

   5. mul0-rgt 0.7%

      \[\leadsto -4 \cdot \frac{\log \color{blue}{0}}{\pi} \]

6. Simplified 0.7%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0}{\pi}} \]

7. Final simplification 0.7%

   \[\leadsto \frac{\log 0}{\pi} \cdot \left(-4\right) \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(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)))))))))