VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 98.9%

Time: 17.8s

Alternatives: 8

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: 6.8% 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: 98.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)}\right) \cdot \frac{-4}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log (+ (/ -1.0 (expm1 (* (* -0.5 PI) f))) (/ 1.0 (expm1 (* f (* PI 0.5))))))
  (/ -4.0 PI)))

C

double code(double f) {
	return log(((-1.0 / expm1(((-0.5 * ((double) M_PI)) * f))) + (1.0 / expm1((f * (((double) M_PI) * 0.5)))))) * (-4.0 / ((double) M_PI));
}

Java

public static double code(double f) {
	return Math.log(((-1.0 / Math.expm1(((-0.5 * Math.PI) * f))) + (1.0 / Math.expm1((f * (Math.PI * 0.5)))))) * (-4.0 / Math.PI);
}

Python

def code(f):
	return math.log(((-1.0 / math.expm1(((-0.5 * math.pi) * f))) + (1.0 / math.expm1((f * (math.pi * 0.5)))))) * (-4.0 / math.pi)

Julia

function code(f)
	return Float64(log(Float64(Float64(-1.0 / expm1(Float64(Float64(-0.5 * pi) * f))) + Float64(1.0 / expm1(Float64(f * Float64(pi * 0.5)))))) * Float64(-4.0 / pi))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(-1.0 / N[(Exp[N[(N[(-0.5 * Pi), $MachinePrecision] * f), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(Exp[N[(f * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

3. Simplified 99.0%

   \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Final simplification 99.0%

   \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)}\right) \cdot \frac{-4}{\pi} \]
6. Add Preprocessing

Alternative 2: 98.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(f \cdot \left(\pi \cdot 0.041666666666666664\right) - 0.5\right) + \frac{2}{\pi}}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{-4}{\pi} \cdot 0}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 225.0)
   (*
    (/ -4.0 PI)
    (log
     (+
      (/ -1.0 (expm1 (* (* -0.5 PI) f)))
      (/ (+ (* f (- (* f (* PI 0.041666666666666664)) 0.5)) (/ 2.0 PI)) f))))
   (pow (cbrt (* (/ -4.0 PI) 0.0)) 3.0)))

C

double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = (-4.0 / ((double) M_PI)) * log(((-1.0 / expm1(((-0.5 * ((double) M_PI)) * f))) + (((f * ((f * (((double) M_PI) * 0.041666666666666664)) - 0.5)) + (2.0 / ((double) M_PI))) / f)));
	} else {
		tmp = pow(cbrt(((-4.0 / ((double) M_PI)) * 0.0)), 3.0);
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = (-4.0 / Math.PI) * Math.log(((-1.0 / Math.expm1(((-0.5 * Math.PI) * f))) + (((f * ((f * (Math.PI * 0.041666666666666664)) - 0.5)) + (2.0 / Math.PI)) / f)));
	} else {
		tmp = Math.pow(Math.cbrt(((-4.0 / Math.PI) * 0.0)), 3.0);
	}
	return tmp;
}

Julia

function code(f)
	tmp = 0.0
	if (f <= 225.0)
		tmp = Float64(Float64(-4.0 / pi) * log(Float64(Float64(-1.0 / expm1(Float64(Float64(-0.5 * pi) * f))) + Float64(Float64(Float64(f * Float64(Float64(f * Float64(pi * 0.041666666666666664)) - 0.5)) + Float64(2.0 / pi)) / f))));
	else
		tmp = cbrt(Float64(Float64(-4.0 / pi) * 0.0)) ^ 3.0;
	end
	return tmp
end

Wolfram

code[f_] := If[LessEqual[f, 225.0], N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(N[(-1.0 / N[(Exp[N[(N[(-0.5 * Pi), $MachinePrecision] * f), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(f * N[(N[(f * N[(Pi * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Pi), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(N[(-4.0 / Pi), $MachinePrecision] * 0.0), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < 225

   1. Initial program 6.9%

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

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
   4. Add Preprocessing

   5. Taylor expanded in f around 0 98.4%

      \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \color{blue}{\frac{f \cdot \left(-1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}}\right) \cdot \frac{-4}{\pi} \]
   6. Step-by-step derivation

      1. pow1 98.4%

         \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{{\left(-1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)}^{1}} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

      2. mul-1-neg 98.4%

         \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left({\color{blue}{\left(-f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)}}^{1} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

      3. distribute-rgt-out 98.4%

         \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left({\left(-f \cdot \color{blue}{\left(\pi \cdot \left(-0.125 + 0.08333333333333333\right)\right)}\right)}^{1} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

      4. metadata-eval 98.4%

         \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left({\left(-f \cdot \left(\pi \cdot \color{blue}{-0.041666666666666664}\right)\right)}^{1} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   7. Applied egg-rr 98.4%

      \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{{\left(-f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)}^{1}} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]
   8. Step-by-step derivation

      1. unpow1 98.4%

         \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{\left(-f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

      2. distribute-rgt-neg-in 98.4%

         \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{f \cdot \left(-\pi \cdot -0.041666666666666664\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

      3. distribute-rgt-neg-in 98.4%

         \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(f \cdot \color{blue}{\left(\pi \cdot \left(--0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

      4. metadata-eval 98.4%

         \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(f \cdot \left(\pi \cdot \color{blue}{0.041666666666666664}\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   9. Simplified 98.4%

      \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{f \cdot \left(\pi \cdot 0.041666666666666664\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]
   10. Step-by-step derivation

       1. un-div-inv 98.4%

          \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(f \cdot \left(\pi \cdot 0.041666666666666664\right) - 0.5\right) + \color{blue}{\frac{2}{\pi}}}{f}\right) \cdot \frac{-4}{\pi} \]

   11. Applied egg-rr 98.4%

       \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(f \cdot \left(\pi \cdot 0.041666666666666664\right) - 0.5\right) + \color{blue}{\frac{2}{\pi}}}{f}\right) \cdot \frac{-4}{\pi} \]

   if 225 < f

   1. Initial program 0.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
   4. Add Preprocessing

   6. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right) \cdot \frac{-4}{\pi}}\right)}^{3}} \]
   7. Step-by-step derivation

      1. flip-+ 0.0%

         \[\leadsto {\left(\sqrt[3]{\log \color{blue}{\left(\frac{\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} \cdot \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} - \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} \cdot \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}}{\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} - \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}}\right)} \cdot \frac{-4}{\pi}}\right)}^{3} \]

      2. log-div 0.0%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} \cdot \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} - \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} \cdot \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right) - \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} - \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right)\right)} \cdot \frac{-4}{\pi}}\right)}^{3} \]

   8. Applied egg-rr 0.0%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(\log \left({\left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right)}^{2} - {\left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right)}^{2}\right) - \log \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} - \frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right)\right)} \cdot \frac{-4}{\pi}}\right)}^{3} \]
   9. Step-by-step derivation

      1. +-inverses 0.0%

         \[\leadsto {\left(\sqrt[3]{\left(\log \color{blue}{0} - \log \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} - \frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right)\right) \cdot \frac{-4}{\pi}}\right)}^{3} \]

      2. +-inverses 0.0%

         \[\leadsto {\left(\sqrt[3]{\left(\log 0 - \log \color{blue}{0}\right) \cdot \frac{-4}{\pi}}\right)}^{3} \]

      3. +-inverses 100.0%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{0} \cdot \frac{-4}{\pi}}\right)}^{3} \]

   10. Simplified 100.0%

       \[\leadsto {\left(\sqrt[3]{\color{blue}{0} \cdot \frac{-4}{\pi}}\right)}^{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(f \cdot \left(\pi \cdot 0.041666666666666664\right) - 0.5\right) + \frac{2}{\pi}}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{-4}{\pi} \cdot 0}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := f \cdot \left(\pi \cdot 0.041666666666666664\right)\\ t_1 := 2 \cdot \frac{1}{\pi}\\ \mathbf{if}\;f \leq 225:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{f \cdot \left(0.5 + t\_0\right) + t\_1}{f} + \frac{f \cdot \left(t\_0 - 0.5\right) + t\_1}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{-4}{\pi} \cdot 0}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* f (* PI 0.041666666666666664))) (t_1 (* 2.0 (/ 1.0 PI))))
   (if (<= f 225.0)
     (*
      (/ -4.0 PI)
      (log
       (+ (/ (+ (* f (+ 0.5 t_0)) t_1) f) (/ (+ (* f (- t_0 0.5)) t_1) f))))
     (pow (cbrt (* (/ -4.0 PI) 0.0)) 3.0))))

C

double code(double f) {
	double t_0 = f * (((double) M_PI) * 0.041666666666666664);
	double t_1 = 2.0 * (1.0 / ((double) M_PI));
	double tmp;
	if (f <= 225.0) {
		tmp = (-4.0 / ((double) M_PI)) * log(((((f * (0.5 + t_0)) + t_1) / f) + (((f * (t_0 - 0.5)) + t_1) / f)));
	} else {
		tmp = pow(cbrt(((-4.0 / ((double) M_PI)) * 0.0)), 3.0);
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = f * (Math.PI * 0.041666666666666664);
	double t_1 = 2.0 * (1.0 / Math.PI);
	double tmp;
	if (f <= 225.0) {
		tmp = (-4.0 / Math.PI) * Math.log(((((f * (0.5 + t_0)) + t_1) / f) + (((f * (t_0 - 0.5)) + t_1) / f)));
	} else {
		tmp = Math.pow(Math.cbrt(((-4.0 / Math.PI) * 0.0)), 3.0);
	}
	return tmp;
}

Julia

function code(f)
	t_0 = Float64(f * Float64(pi * 0.041666666666666664))
	t_1 = Float64(2.0 * Float64(1.0 / pi))
	tmp = 0.0
	if (f <= 225.0)
		tmp = Float64(Float64(-4.0 / pi) * log(Float64(Float64(Float64(Float64(f * Float64(0.5 + t_0)) + t_1) / f) + Float64(Float64(Float64(f * Float64(t_0 - 0.5)) + t_1) / f))));
	else
		tmp = cbrt(Float64(Float64(-4.0 / pi) * 0.0)) ^ 3.0;
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[(f * N[(Pi * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[f, 225.0], N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(N[(N[(N[(f * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] / f), $MachinePrecision] + N[(N[(N[(f * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(N[(-4.0 / Pi), $MachinePrecision] * 0.0), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if f < 225

   1. Initial program 6.9%

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

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
   4. Add Preprocessing

   5. Taylor expanded in f around 0 98.4%

      \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \color{blue}{\frac{f \cdot \left(-1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}}\right) \cdot \frac{-4}{\pi} \]
   6. Step-by-step derivation

      1. pow1 98.4%

         \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{{\left(-1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)}^{1}} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

      2. mul-1-neg 98.4%

         \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left({\color{blue}{\left(-f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)}}^{1} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

      3. distribute-rgt-out 98.4%

         \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left({\left(-f \cdot \color{blue}{\left(\pi \cdot \left(-0.125 + 0.08333333333333333\right)\right)}\right)}^{1} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

      4. metadata-eval 98.4%

         \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left({\left(-f \cdot \left(\pi \cdot \color{blue}{-0.041666666666666664}\right)\right)}^{1} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   7. Applied egg-rr 98.4%

      \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{{\left(-f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)}^{1}} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]
   8. Step-by-step derivation

      1. unpow1 98.4%

         \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{\left(-f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

      2. distribute-rgt-neg-in 98.4%

         \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{f \cdot \left(-\pi \cdot -0.041666666666666664\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

      3. distribute-rgt-neg-in 98.4%

         \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(f \cdot \color{blue}{\left(\pi \cdot \left(--0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

      4. metadata-eval 98.4%

         \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(f \cdot \left(\pi \cdot \color{blue}{0.041666666666666664}\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   9. Simplified 98.4%

      \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{f \cdot \left(\pi \cdot 0.041666666666666664\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   10. Taylor expanded in f around 0 98.3%

       \[\leadsto \log \left(\color{blue}{\frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}} + \frac{f \cdot \left(f \cdot \left(\pi \cdot 0.041666666666666664\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]
   11. Step-by-step derivation

       1. pow1 98.4%

          \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{{\left(-1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)}^{1}} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

       2. mul-1-neg 98.4%

          \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left({\color{blue}{\left(-f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)}}^{1} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

       3. distribute-rgt-out 98.4%

          \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left({\left(-f \cdot \color{blue}{\left(\pi \cdot \left(-0.125 + 0.08333333333333333\right)\right)}\right)}^{1} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

       4. metadata-eval 98.4%

          \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left({\left(-f \cdot \left(\pi \cdot \color{blue}{-0.041666666666666664}\right)\right)}^{1} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   12. Applied egg-rr 98.3%

       \[\leadsto \log \left(\frac{f \cdot \left(0.5 + \color{blue}{{\left(-f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)}^{1}}\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(f \cdot \left(\pi \cdot 0.041666666666666664\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]
   13. Step-by-step derivation

       1. unpow1 98.4%

          \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{\left(-f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

       2. distribute-rgt-neg-in 98.4%

          \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{f \cdot \left(-\pi \cdot -0.041666666666666664\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

       3. distribute-rgt-neg-in 98.4%

          \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(f \cdot \color{blue}{\left(\pi \cdot \left(--0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

       4. metadata-eval 98.4%

          \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(f \cdot \left(\pi \cdot \color{blue}{0.041666666666666664}\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   14. Simplified 98.3%

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

   if 225 < f

   1. Initial program 0.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
   4. Add Preprocessing

   6. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right) \cdot \frac{-4}{\pi}}\right)}^{3}} \]
   7. Step-by-step derivation

      1. flip-+ 0.0%

         \[\leadsto {\left(\sqrt[3]{\log \color{blue}{\left(\frac{\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} \cdot \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} - \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} \cdot \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}}{\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} - \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}}\right)} \cdot \frac{-4}{\pi}}\right)}^{3} \]

      2. log-div 0.0%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} \cdot \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} - \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} \cdot \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right) - \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} - \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right)\right)} \cdot \frac{-4}{\pi}}\right)}^{3} \]

   8. Applied egg-rr 0.0%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(\log \left({\left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right)}^{2} - {\left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right)}^{2}\right) - \log \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} - \frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right)\right)} \cdot \frac{-4}{\pi}}\right)}^{3} \]
   9. Step-by-step derivation

      1. +-inverses 0.0%

         \[\leadsto {\left(\sqrt[3]{\left(\log \color{blue}{0} - \log \left(\frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)} - \frac{-1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot -0.5\right)\right)}\right)\right) \cdot \frac{-4}{\pi}}\right)}^{3} \]

      2. +-inverses 0.0%

         \[\leadsto {\left(\sqrt[3]{\left(\log 0 - \log \color{blue}{0}\right) \cdot \frac{-4}{\pi}}\right)}^{3} \]

      3. +-inverses 100.0%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{0} \cdot \frac{-4}{\pi}}\right)}^{3} \]

   10. Simplified 100.0%

       \[\leadsto {\left(\sqrt[3]{\color{blue}{0} \cdot \frac{-4}{\pi}}\right)}^{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{f \cdot \left(0.5 + f \cdot \left(\pi \cdot 0.041666666666666664\right)\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(f \cdot \left(\pi \cdot 0.041666666666666664\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{-4}{\pi} \cdot 0}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \frac{1}{\pi}\\ t_1 := f \cdot \left(\pi \cdot 0.041666666666666664\right)\\ \frac{-4}{\pi} \cdot \log \left(\frac{f \cdot \left(0.5 + t\_1\right) + t\_0}{f} + \frac{f \cdot \left(t\_1 - 0.5\right) + t\_0}{f}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* 2.0 (/ 1.0 PI))) (t_1 (* f (* PI 0.041666666666666664))))
   (*
    (/ -4.0 PI)
    (log
     (+ (/ (+ (* f (+ 0.5 t_1)) t_0) f) (/ (+ (* f (- t_1 0.5)) t_0) f))))))

C

double code(double f) {
	double t_0 = 2.0 * (1.0 / ((double) M_PI));
	double t_1 = f * (((double) M_PI) * 0.041666666666666664);
	return (-4.0 / ((double) M_PI)) * log(((((f * (0.5 + t_1)) + t_0) / f) + (((f * (t_1 - 0.5)) + t_0) / f)));
}

Java

public static double code(double f) {
	double t_0 = 2.0 * (1.0 / Math.PI);
	double t_1 = f * (Math.PI * 0.041666666666666664);
	return (-4.0 / Math.PI) * Math.log(((((f * (0.5 + t_1)) + t_0) / f) + (((f * (t_1 - 0.5)) + t_0) / f)));
}

Python

def code(f):
	t_0 = 2.0 * (1.0 / math.pi)
	t_1 = f * (math.pi * 0.041666666666666664)
	return (-4.0 / math.pi) * math.log(((((f * (0.5 + t_1)) + t_0) / f) + (((f * (t_1 - 0.5)) + t_0) / f)))

Julia

function code(f)
	t_0 = Float64(2.0 * Float64(1.0 / pi))
	t_1 = Float64(f * Float64(pi * 0.041666666666666664))
	return Float64(Float64(-4.0 / pi) * log(Float64(Float64(Float64(Float64(f * Float64(0.5 + t_1)) + t_0) / f) + Float64(Float64(Float64(f * Float64(t_1 - 0.5)) + t_0) / f))))
end

MATLAB

function tmp = code(f)
	t_0 = 2.0 * (1.0 / pi);
	t_1 = f * (pi * 0.041666666666666664);
	tmp = (-4.0 / pi) * log(((((f * (0.5 + t_1)) + t_0) / f) + (((f * (t_1 - 0.5)) + t_0) / f)));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(f * N[(Pi * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(N[(N[(N[(f * N[(0.5 + t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] / f), $MachinePrecision] + N[(N[(N[(f * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 6.8%

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

3. Simplified 99.0%

   \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 96.1%

   \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \color{blue}{\frac{f \cdot \left(-1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}}\right) \cdot \frac{-4}{\pi} \]
6. Step-by-step derivation

   1. pow1 96.1%

      \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{{\left(-1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)}^{1}} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   2. mul-1-neg 96.1%

      \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left({\color{blue}{\left(-f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)}}^{1} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   3. distribute-rgt-out 96.1%

      \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left({\left(-f \cdot \color{blue}{\left(\pi \cdot \left(-0.125 + 0.08333333333333333\right)\right)}\right)}^{1} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   4. metadata-eval 96.1%

      \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left({\left(-f \cdot \left(\pi \cdot \color{blue}{-0.041666666666666664}\right)\right)}^{1} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

7. Applied egg-rr 96.1%

   \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{{\left(-f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)}^{1}} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]
8. Step-by-step derivation

   1. unpow1 96.1%

      \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{\left(-f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   2. distribute-rgt-neg-in 96.1%

      \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{f \cdot \left(-\pi \cdot -0.041666666666666664\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   3. distribute-rgt-neg-in 96.1%

      \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(f \cdot \color{blue}{\left(\pi \cdot \left(--0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   4. metadata-eval 96.1%

      \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(f \cdot \left(\pi \cdot \color{blue}{0.041666666666666664}\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

9. Simplified 96.1%

   \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{f \cdot \left(\pi \cdot 0.041666666666666664\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

10. Taylor expanded in f around 0 96.1%

    \[\leadsto \log \left(\color{blue}{\frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}} + \frac{f \cdot \left(f \cdot \left(\pi \cdot 0.041666666666666664\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]
11. Step-by-step derivation

    1. pow1 96.1%

       \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{{\left(-1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)}^{1}} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

    2. mul-1-neg 96.1%

       \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left({\color{blue}{\left(-f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)}}^{1} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

    3. distribute-rgt-out 96.1%

       \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left({\left(-f \cdot \color{blue}{\left(\pi \cdot \left(-0.125 + 0.08333333333333333\right)\right)}\right)}^{1} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

    4. metadata-eval 96.1%

       \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left({\left(-f \cdot \left(\pi \cdot \color{blue}{-0.041666666666666664}\right)\right)}^{1} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

12. Applied egg-rr 96.1%

    \[\leadsto \log \left(\frac{f \cdot \left(0.5 + \color{blue}{{\left(-f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)}^{1}}\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(f \cdot \left(\pi \cdot 0.041666666666666664\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]
13. Step-by-step derivation

    1. unpow1 96.1%

       \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{\left(-f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

    2. distribute-rgt-neg-in 96.1%

       \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(\color{blue}{f \cdot \left(-\pi \cdot -0.041666666666666664\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

    3. distribute-rgt-neg-in 96.1%

       \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(f \cdot \color{blue}{\left(\pi \cdot \left(--0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

    4. metadata-eval 96.1%

       \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{f \cdot \left(f \cdot \left(\pi \cdot \color{blue}{0.041666666666666664}\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

14. Simplified 96.1%

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

15. Final simplification 96.1%

    \[\leadsto \frac{-4}{\pi} \cdot \log \left(\frac{f \cdot \left(0.5 + f \cdot \left(\pi \cdot 0.041666666666666664\right)\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(f \cdot \left(\pi \cdot 0.041666666666666664\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \]
16. Add Preprocessing

Alternative 5: 96.0% accurate, 4.9× speedup

Math

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

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(Float64(-4.0 * log(Float64(Float64(4.0 / pi) / f))) / pi)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

3. Simplified 99.0%

   \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 95.5%

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

   1. mul-1-neg 95.5%

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

   2. unsub-neg 95.5%

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

7. Simplified 95.5%

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

   1. associate-*r/ 95.5%

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

   2. diff-log 95.4%

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

9. Applied egg-rr 95.4%

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

10. Final simplification 95.4%

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

Alternative 6: 95.9% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

3. Simplified 99.0%

   \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 95.5%

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

   1. mul-1-neg 95.5%

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

   2. unsub-neg 95.5%

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

7. Simplified 95.5%

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

   1. associate-*r/ 95.5%

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

   2. diff-log 95.4%

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

9. Applied egg-rr 95.4%

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

    1. associate-/l* 95.3%

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

11. Simplified 95.3%

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

12. Final simplification 95.3%

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

Alternative 7: 95.9% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

3. Simplified 99.0%

   \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 95.3%

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

   1. *-commutative 95.3%

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

7. Simplified 95.3%

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

8. Final simplification 95.3%

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

Alternative 8: 3.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \log 0 \end{array} \]

FPCore

(FPCore (f) :precision binary64 (log 0.0))

C

double code(double f) {
	return log(0.0);
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = log(0.0d0)
end function

Java

public static double code(double f) {
	return Math.log(0.0);
}

Python

def code(f):
	return math.log(0.0)

Julia

function code(f)
	return log(0.0)
end

MATLAB

function tmp = code(f)
	tmp = log(0.0);
end

Wolfram

code[f_] := N[Log[0.0], $MachinePrecision]

Derivation

1. Initial program 6.8%

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

3. Simplified 99.0%

   \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

6. Applied egg-rr 0.7%

   \[\leadsto \log \color{blue}{\left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} - \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right)} \cdot \frac{-4}{\pi} \]
7. Step-by-step derivation

   1. +-inverses 0.7%

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

8. Simplified 0.7%

   \[\leadsto \log \color{blue}{0} \cdot \frac{-4}{\pi} \]
9. Step-by-step derivation

   1. add-log-exp 0.7%

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

   2. exp-to-pow 0.7%

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

10. Applied egg-rr 0.7%

    \[\leadsto \color{blue}{\log \left({0}^{\left(\frac{-4}{\pi}\right)}\right)} \]
11. Step-by-step derivation

    1. pow-base-0 3.1%

       \[\leadsto \log \color{blue}{0} \]

12. Simplified 3.1%

    \[\leadsto \color{blue}{\log 0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024159 
(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)))))))))