VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 98.2%

Time: 46.1s

Alternatives: 9

Speedup: 3.3×

• could not determine a ground truth

  1. f = -2.387213994773681e+52

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: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\pi \cdot 0.5\right)}^{2}\\ \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 200:\\ \;\;\;\;-\mathsf{fma}\left(2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right), \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\frac{t_0}{0.005208333333333333}} \cdot -2\right), 0 \cdot t_0\right), \frac{4}{\frac{\pi}{\log \left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{\frac{\pi}{0}} \cdot \left(-2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (pow (* PI 0.5) 2.0)))
   (if (<= (* (/ PI 4.0) f) 200.0)
     (-
      (fma
       2.0
       (* (/ f PI) (* PI 0.0))
       (fma
        2.0
        (*
         (/ (pow f 2.0) PI)
         (fma
          (* PI 0.5)
          (fma
           0.0625
           (/ (pow PI 2.0) (* PI 0.5))
           (* (/ (pow PI 3.0) (/ t_0 0.005208333333333333)) -2.0))
          (* 0.0 t_0)))
        (/ 4.0 (/ PI (log (/ (/ 2.0 (* PI 0.5)) f)))))))
     (* (/ f (/ PI 0.0)) (- 2.0)))))

C

double code(double f) {
	double t_0 = pow((((double) M_PI) * 0.5), 2.0);
	double tmp;
	if (((((double) M_PI) / 4.0) * f) <= 200.0) {
		tmp = -fma(2.0, ((f / ((double) M_PI)) * (((double) M_PI) * 0.0)), fma(2.0, ((pow(f, 2.0) / ((double) M_PI)) * fma((((double) M_PI) * 0.5), fma(0.0625, (pow(((double) M_PI), 2.0) / (((double) M_PI) * 0.5)), ((pow(((double) M_PI), 3.0) / (t_0 / 0.005208333333333333)) * -2.0)), (0.0 * t_0))), (4.0 / (((double) M_PI) / log(((2.0 / (((double) M_PI) * 0.5)) / f))))));
	} else {
		tmp = (f / (((double) M_PI) / 0.0)) * -2.0;
	}
	return tmp;
}

Julia

function code(f)
	t_0 = Float64(pi * 0.5) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(pi / 4.0) * f) <= 200.0)
		tmp = Float64(-fma(2.0, Float64(Float64(f / pi) * Float64(pi * 0.0)), fma(2.0, Float64(Float64((f ^ 2.0) / pi) * fma(Float64(pi * 0.5), fma(0.0625, Float64((pi ^ 2.0) / Float64(pi * 0.5)), Float64(Float64((pi ^ 3.0) / Float64(t_0 / 0.005208333333333333)) * -2.0)), Float64(0.0 * t_0))), Float64(4.0 / Float64(pi / log(Float64(Float64(2.0 / Float64(pi * 0.5)) / f)))))));
	else
		tmp = Float64(Float64(f / Float64(pi / 0.0)) * Float64(-2.0));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[Power[N[(Pi * 0.5), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision], 200.0], (-N[(2.0 * N[(N[(f / Pi), $MachinePrecision] * N[(Pi * 0.0), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(N[Power[f, 2.0], $MachinePrecision] / Pi), $MachinePrecision] * N[(N[(Pi * 0.5), $MachinePrecision] * N[(0.0625 * N[(N[Power[Pi, 2.0], $MachinePrecision] / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[Pi, 3.0], $MachinePrecision] / N[(t$95$0 / 0.005208333333333333), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + N[(0.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 / N[(Pi / N[Log[N[(N[(2.0 / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(f / N[(Pi / 0.0), $MachinePrecision]), $MachinePrecision] * (-2.0)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) 4) f) < 200

   1. Initial program 6.1%

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

   2. Taylor expanded in f around 0 98.3%

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

   4. Simplified 98.2%

      \[\leadsto -\color{blue}{\mathsf{fma}\left(2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right), \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right), \frac{4}{\frac{\pi}{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}}\right)\right)} \]
   5. Step-by-step derivation

      1. diff-log 98.3%

         \[\leadsto -\mathsf{fma}\left(2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right), \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right), \frac{4}{\frac{\pi}{\color{blue}{\log \left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)}}}\right)\right) \]

   6. Applied egg-rr 98.3%

      \[\leadsto -\mathsf{fma}\left(2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right), \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right), \frac{4}{\frac{\pi}{\color{blue}{\log \left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)}}}\right)\right) \]

   if 200 < (*.f64 (/.f64 (PI.f64) 4) 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. Applied egg-rr 1.6%

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

   4. Taylor expanded in f around 0 100.0%

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

      1. associate-/l* 100.0%

         \[\leadsto -2 \cdot \color{blue}{\frac{f}{\frac{\pi}{-0.25 \cdot \pi + 0.25 \cdot \pi}}} \]

      2. distribute-rgt-out 100.0%

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

      3. metadata-eval 100.0%

         \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\pi \cdot \color{blue}{0}}} \]

      4. mul0-rgt 100.0%

         \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\color{blue}{0}}} \]

   6. Simplified 100.0%

      \[\leadsto -\color{blue}{2 \cdot \frac{f}{\frac{\pi}{0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 200:\\ \;\;\;\;-\mathsf{fma}\left(2, \frac{f}{\pi} \cdot \left(\pi \cdot 0\right), \mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2\right), 0 \cdot {\left(\pi \cdot 0.5\right)}^{2}\right), \frac{4}{\frac{\pi}{\log \left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{\frac{\pi}{0}} \cdot \left(-2\right)\\ \end{array} \]

Alternative 2: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 200:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot \left(\log f - \mathsf{fma}\left({f}^{2}, 0.5 \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), 0\right), \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{\frac{\pi}{0}} \cdot \left(-2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= (* (/ PI 4.0) f) 200.0)
   (*
    (/ 1.0 (/ PI 4.0))
    (-
     (log f)
     (fma
      (pow f 2.0)
      (*
       0.5
       (fma
        PI
        (* 0.5 (fma (* PI 0.020833333333333332) -2.0 (* 0.0625 (/ PI 0.5))))
        0.0))
      (log (/ (/ 2.0 PI) 0.5)))))
   (* (/ f (/ PI 0.0)) (- 2.0))))

C

double code(double f) {
	double tmp;
	if (((((double) M_PI) / 4.0) * f) <= 200.0) {
		tmp = (1.0 / (((double) M_PI) / 4.0)) * (log(f) - fma(pow(f, 2.0), (0.5 * fma(((double) M_PI), (0.5 * fma((((double) M_PI) * 0.020833333333333332), -2.0, (0.0625 * (((double) M_PI) / 0.5)))), 0.0)), log(((2.0 / ((double) M_PI)) / 0.5))));
	} else {
		tmp = (f / (((double) M_PI) / 0.0)) * -2.0;
	}
	return tmp;
}

Julia

function code(f)
	tmp = 0.0
	if (Float64(Float64(pi / 4.0) * f) <= 200.0)
		tmp = Float64(Float64(1.0 / Float64(pi / 4.0)) * Float64(log(f) - fma((f ^ 2.0), Float64(0.5 * fma(pi, Float64(0.5 * fma(Float64(pi * 0.020833333333333332), -2.0, Float64(0.0625 * Float64(pi / 0.5)))), 0.0)), log(Float64(Float64(2.0 / pi) / 0.5)))));
	else
		tmp = Float64(Float64(f / Float64(pi / 0.0)) * Float64(-2.0));
	end
	return tmp
end

Wolfram

code[f_] := If[LessEqual[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision], 200.0], N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[Log[f], $MachinePrecision] - N[(N[Power[f, 2.0], $MachinePrecision] * N[(0.5 * N[(Pi * N[(0.5 * N[(N[(Pi * 0.020833333333333332), $MachinePrecision] * -2.0 + N[(0.0625 * N[(Pi / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision] + N[Log[N[(N[(2.0 / Pi), $MachinePrecision] / 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(f / N[(Pi / 0.0), $MachinePrecision]), $MachinePrecision] * (-2.0)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) 4) f) < 200

   1. Initial program 6.1%

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

   2. Taylor expanded in f around 0 98.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \left(-1 \cdot \log f + \left(0.5 \cdot \left(f \cdot \left(\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot \left({\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2} \cdot {\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}\right) + \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right) \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right)\right)\right)\right)} \]

   4. Simplified 98.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\mathsf{fma}\left({f}^{2}, \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), 0\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 98.2%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left({f}^{2}, \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right)\right)}, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), 0\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

      2. expm1-udef 98.2%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left({f}^{2}, \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\color{blue}{e^{\mathsf{log1p}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right)} - 1}, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), 0\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

      3. pow-div 98.2%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left({f}^{2}, \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{\left(3 - 2\right)}} \cdot 0.020833333333333332\right)} - 1, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), 0\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

      4. metadata-eval 98.2%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left({f}^{2}, \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(e^{\mathsf{log1p}\left({\pi}^{\color{blue}{1}} \cdot 0.020833333333333332\right)} - 1, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), 0\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

      5. pow1 98.2%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left({f}^{2}, \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(e^{\mathsf{log1p}\left(\color{blue}{\pi} \cdot 0.020833333333333332\right)} - 1, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), 0\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

   6. Applied egg-rr 98.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left({f}^{2}, \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\color{blue}{e^{\mathsf{log1p}\left(\pi \cdot 0.020833333333333332\right)} - 1}, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), 0\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]
   7. Step-by-step derivation

      1. expm1-def 98.2%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\mathsf{fma}\left({f}^{2}, \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot 0.020833333333333332\right)\right)}, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), 0\right) \cdot 0.5, \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right) - \log f\right) \]

      2. expm1-log1p 98.2%

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

   8. Simplified 98.2%

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

   if 200 < (*.f64 (/.f64 (PI.f64) 4) 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. Applied egg-rr 1.6%

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

   4. Taylor expanded in f around 0 100.0%

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

      1. associate-/l* 100.0%

         \[\leadsto -2 \cdot \color{blue}{\frac{f}{\frac{\pi}{-0.25 \cdot \pi + 0.25 \cdot \pi}}} \]

      2. distribute-rgt-out 100.0%

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

      3. metadata-eval 100.0%

         \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\pi \cdot \color{blue}{0}}} \]

      4. mul0-rgt 100.0%

         \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\color{blue}{0}}} \]

   6. Simplified 100.0%

      \[\leadsto -\color{blue}{2 \cdot \frac{f}{\frac{\pi}{0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 200:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot \left(\log f - \mathsf{fma}\left({f}^{2}, 0.5 \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), 0\right), \log \left(\frac{\frac{2}{\pi}}{0.5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{\frac{\pi}{0}} \cdot \left(-2\right)\\ \end{array} \]

Alternative 3: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 200:\\ \;\;\;\;\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.020833333333333332 \cdot \frac{{\pi}^{3}}{{\pi}^{2}}, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), 4 \cdot \frac{\frac{1}{f}}{\pi}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{\frac{\pi}{0}} \cdot \left(-2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= (* (/ PI 4.0) f) 200.0)
   (*
    (log
     (fma
      f
      (fma
       (* 0.020833333333333332 (/ (pow PI 3.0) (pow PI 2.0)))
       -2.0
       (* 0.0625 (/ PI 0.5)))
      (* 4.0 (/ (/ 1.0 f) PI))))
    (/ -1.0 (/ PI 4.0)))
   (* (/ f (/ PI 0.0)) (- 2.0))))

C

double code(double f) {
	double tmp;
	if (((((double) M_PI) / 4.0) * f) <= 200.0) {
		tmp = log(fma(f, fma((0.020833333333333332 * (pow(((double) M_PI), 3.0) / pow(((double) M_PI), 2.0))), -2.0, (0.0625 * (((double) M_PI) / 0.5))), (4.0 * ((1.0 / f) / ((double) M_PI))))) * (-1.0 / (((double) M_PI) / 4.0));
	} else {
		tmp = (f / (((double) M_PI) / 0.0)) * -2.0;
	}
	return tmp;
}

Julia

function code(f)
	tmp = 0.0
	if (Float64(Float64(pi / 4.0) * f) <= 200.0)
		tmp = Float64(log(fma(f, fma(Float64(0.020833333333333332 * Float64((pi ^ 3.0) / (pi ^ 2.0))), -2.0, Float64(0.0625 * Float64(pi / 0.5))), Float64(4.0 * Float64(Float64(1.0 / f) / pi)))) * Float64(-1.0 / Float64(pi / 4.0)));
	else
		tmp = Float64(Float64(f / Float64(pi / 0.0)) * Float64(-2.0));
	end
	return tmp
end

Wolfram

code[f_] := If[LessEqual[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision], 200.0], N[(N[Log[N[(f * N[(N[(0.020833333333333332 * N[(N[Power[Pi, 3.0], $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + N[(0.0625 * N[(Pi / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(1.0 / f), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(f / N[(Pi / 0.0), $MachinePrecision]), $MachinePrecision] * (-2.0)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) 4) f) < 200

   1. Initial program 6.1%

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

   2. Taylor expanded in f around 0 98.2%

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

   4. Simplified 98.2%

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

   if 200 < (*.f64 (/.f64 (PI.f64) 4) 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. Applied egg-rr 1.6%

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

   4. Taylor expanded in f around 0 100.0%

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

      1. associate-/l* 100.0%

         \[\leadsto -2 \cdot \color{blue}{\frac{f}{\frac{\pi}{-0.25 \cdot \pi + 0.25 \cdot \pi}}} \]

      2. distribute-rgt-out 100.0%

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

      3. metadata-eval 100.0%

         \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\pi \cdot \color{blue}{0}}} \]

      4. mul0-rgt 100.0%

         \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\color{blue}{0}}} \]

   6. Simplified 100.0%

      \[\leadsto -\color{blue}{2 \cdot \frac{f}{\frac{\pi}{0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 200:\\ \;\;\;\;\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.020833333333333332 \cdot \frac{{\pi}^{3}}{{\pi}^{2}}, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), 4 \cdot \frac{\frac{1}{f}}{\pi}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{\frac{\pi}{0}} \cdot \left(-2\right)\\ \end{array} \]

Alternative 4: 98.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 200:\\ \;\;\;\;\frac{-\log \left(2 \cdot \frac{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{f \cdot \left(\pi \cdot 0.5\right)}\right)}{\pi \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{\frac{\pi}{0}} \cdot \left(-2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= (* (/ PI 4.0) f) 200.0)
   (/
    (- (log (* 2.0 (/ (cosh (* f (* PI 0.25))) (* f (* PI 0.5))))))
    (* PI 0.25))
   (* (/ f (/ PI 0.0)) (- 2.0))))

C

double code(double f) {
	double tmp;
	if (((((double) M_PI) / 4.0) * f) <= 200.0) {
		tmp = -log((2.0 * (cosh((f * (((double) M_PI) * 0.25))) / (f * (((double) M_PI) * 0.5))))) / (((double) M_PI) * 0.25);
	} else {
		tmp = (f / (((double) M_PI) / 0.0)) * -2.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (((Math.PI / 4.0) * f) <= 200.0) {
		tmp = -Math.log((2.0 * (Math.cosh((f * (Math.PI * 0.25))) / (f * (Math.PI * 0.5))))) / (Math.PI * 0.25);
	} else {
		tmp = (f / (Math.PI / 0.0)) * -2.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if ((math.pi / 4.0) * f) <= 200.0:
		tmp = -math.log((2.0 * (math.cosh((f * (math.pi * 0.25))) / (f * (math.pi * 0.5))))) / (math.pi * 0.25)
	else:
		tmp = (f / (math.pi / 0.0)) * -2.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (Float64(Float64(pi / 4.0) * f) <= 200.0)
		tmp = Float64(Float64(-log(Float64(2.0 * Float64(cosh(Float64(f * Float64(pi * 0.25))) / Float64(f * Float64(pi * 0.5)))))) / Float64(pi * 0.25));
	else
		tmp = Float64(Float64(f / Float64(pi / 0.0)) * Float64(-2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (((pi / 4.0) * f) <= 200.0)
		tmp = -log((2.0 * (cosh((f * (pi * 0.25))) / (f * (pi * 0.5))))) / (pi * 0.25);
	else
		tmp = (f / (pi / 0.0)) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision], 200.0], N[((-N[Log[N[(2.0 * N[(N[Cosh[N[(f * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(f * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision], N[(N[(f / N[(Pi / 0.0), $MachinePrecision]), $MachinePrecision] * (-2.0)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) 4) f) < 200

   1. Initial program 6.1%

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

   2. Taylor expanded in f around 0 97.5%

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

      1. distribute-rgt-out-- 97.5%

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

      2. metadata-eval 97.5%

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

   4. Simplified 97.5%

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

      1. associate-*l/ 97.7%

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

   6. Applied egg-rr 97.7%

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

   if 200 < (*.f64 (/.f64 (PI.f64) 4) 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. Applied egg-rr 1.6%

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

   4. Taylor expanded in f around 0 100.0%

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

      1. associate-/l* 100.0%

         \[\leadsto -2 \cdot \color{blue}{\frac{f}{\frac{\pi}{-0.25 \cdot \pi + 0.25 \cdot \pi}}} \]

      2. distribute-rgt-out 100.0%

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

      3. metadata-eval 100.0%

         \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\pi \cdot \color{blue}{0}}} \]

      4. mul0-rgt 100.0%

         \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\color{blue}{0}}} \]

   6. Simplified 100.0%

      \[\leadsto -\color{blue}{2 \cdot \frac{f}{\frac{\pi}{0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 200:\\ \;\;\;\;\frac{-\log \left(2 \cdot \frac{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{f \cdot \left(\pi \cdot 0.5\right)}\right)}{\pi \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{\frac{\pi}{0}} \cdot \left(-2\right)\\ \end{array} \]

Alternative 5: 97.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 230:\\ \;\;\;\;-\left|4 \cdot \frac{\log \left(\frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)}{\pi}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{\frac{\pi}{0}} \cdot \left(-2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 230.0)
   (- (fabs (* 4.0 (/ (log (/ 2.0 (* PI (* f 0.5)))) PI))))
   (* (/ f (/ PI 0.0)) (- 2.0))))

C

double code(double f) {
	double tmp;
	if (f <= 230.0) {
		tmp = -fabs((4.0 * (log((2.0 / (((double) M_PI) * (f * 0.5)))) / ((double) M_PI))));
	} else {
		tmp = (f / (((double) M_PI) / 0.0)) * -2.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 230.0) {
		tmp = -Math.abs((4.0 * (Math.log((2.0 / (Math.PI * (f * 0.5)))) / Math.PI)));
	} else {
		tmp = (f / (Math.PI / 0.0)) * -2.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 230.0:
		tmp = -math.fabs((4.0 * (math.log((2.0 / (math.pi * (f * 0.5)))) / math.pi)))
	else:
		tmp = (f / (math.pi / 0.0)) * -2.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 230.0)
		tmp = Float64(-abs(Float64(4.0 * Float64(log(Float64(2.0 / Float64(pi * Float64(f * 0.5)))) / pi))));
	else
		tmp = Float64(Float64(f / Float64(pi / 0.0)) * Float64(-2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 230.0)
		tmp = -abs((4.0 * (log((2.0 / (pi * (f * 0.5)))) / pi)));
	else
		tmp = (f / (pi / 0.0)) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 230.0], (-N[Abs[N[(4.0 * N[(N[Log[N[(2.0 / N[(Pi * N[(f * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(f / N[(Pi / 0.0), $MachinePrecision]), $MachinePrecision] * (-2.0)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < 230

   1. Initial program 6.1%

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

   2. Taylor expanded in f around 0 97.5%

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

      1. associate-/r* 97.5%

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

      2. distribute-rgt-out-- 97.5%

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

      3. metadata-eval 97.5%

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

   4. Simplified 97.5%

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

      1. add-sqr-sqrt 97.1%

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

      2. sqrt-unprod 97.5%

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

      3. pow2 97.5%

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

      4. associate-*l/ 97.7%

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

      5. *-un-lft-identity 97.7%

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

      6. associate-/l/ 97.7%

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

      7. div-inv 97.7%

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

      8. metadata-eval 97.7%

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

   6. Applied egg-rr 97.7%

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

      1. unpow2 97.7%

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

      2. rem-sqrt-square 97.7%

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

      3. associate-/r* 97.7%

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

      4. *-lft-identity 97.7%

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

      5. *-commutative 97.7%

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

      6. times-frac 97.7%

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

      7. metadata-eval 97.7%

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

      8. associate-/r* 97.7%

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

      9. associate-*l* 97.7%

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

   8. Simplified 97.7%

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

   if 230 < 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. Applied egg-rr 1.6%

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

   4. Taylor expanded in f around 0 100.0%

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

      1. associate-/l* 100.0%

         \[\leadsto -2 \cdot \color{blue}{\frac{f}{\frac{\pi}{-0.25 \cdot \pi + 0.25 \cdot \pi}}} \]

      2. distribute-rgt-out 100.0%

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

      3. metadata-eval 100.0%

         \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\pi \cdot \color{blue}{0}}} \]

      4. mul0-rgt 100.0%

         \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\color{blue}{0}}} \]

   6. Simplified 100.0%

      \[\leadsto -\color{blue}{2 \cdot \frac{f}{\frac{\pi}{0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 230:\\ \;\;\;\;-\left|4 \cdot \frac{\log \left(\frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)}{\pi}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{\frac{\pi}{0}} \cdot \left(-2\right)\\ \end{array} \]

Alternative 6: 97.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.25:\\ \;\;\;\;\frac{-\log \left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right)}{\pi \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{\frac{\pi}{0}} \cdot \left(-2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.25)
   (/ (- (log (/ 2.0 (* f (* PI 0.5))))) (* PI 0.25))
   (* (/ f (/ PI 0.0)) (- 2.0))))

C

double code(double f) {
	double tmp;
	if (f <= 1.25) {
		tmp = -log((2.0 / (f * (((double) M_PI) * 0.5)))) / (((double) M_PI) * 0.25);
	} else {
		tmp = (f / (((double) M_PI) / 0.0)) * -2.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.25) {
		tmp = -Math.log((2.0 / (f * (Math.PI * 0.5)))) / (Math.PI * 0.25);
	} else {
		tmp = (f / (Math.PI / 0.0)) * -2.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.25:
		tmp = -math.log((2.0 / (f * (math.pi * 0.5)))) / (math.pi * 0.25)
	else:
		tmp = (f / (math.pi / 0.0)) * -2.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.25)
		tmp = Float64(Float64(-log(Float64(2.0 / Float64(f * Float64(pi * 0.5))))) / Float64(pi * 0.25));
	else
		tmp = Float64(Float64(f / Float64(pi / 0.0)) * Float64(-2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.25)
		tmp = -log((2.0 / (f * (pi * 0.5)))) / (pi * 0.25);
	else
		tmp = (f / (pi / 0.0)) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.25], N[((-N[Log[N[(2.0 / N[(f * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision], N[(N[(f / N[(Pi / 0.0), $MachinePrecision]), $MachinePrecision] * (-2.0)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < 1.25

   1. Initial program 6.1%

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

   2. Taylor expanded in f around 0 98.2%

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

      1. associate-/r* 98.2%

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

      2. distribute-rgt-out-- 98.2%

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

      3. metadata-eval 98.2%

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

   4. Simplified 98.2%

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

      1. associate-*l/ 98.4%

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

      2. *-un-lft-identity 98.4%

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

      3. associate-/l/ 98.4%

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

      4. div-inv 98.4%

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

      5. metadata-eval 98.4%

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

   6. Applied egg-rr 98.4%

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

   if 1.25 < f

   1. Initial program 0.9%

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

   3. Applied egg-rr 2.8%

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

   4. Taylor expanded in f around 0 75.9%

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

      1. associate-/l* 75.9%

         \[\leadsto -2 \cdot \color{blue}{\frac{f}{\frac{\pi}{-0.25 \cdot \pi + 0.25 \cdot \pi}}} \]

      2. distribute-rgt-out 75.9%

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

      3. metadata-eval 75.9%

         \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\pi \cdot \color{blue}{0}}} \]

      4. mul0-rgt 75.9%

         \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\color{blue}{0}}} \]

   6. Simplified 75.9%

      \[\leadsto -\color{blue}{2 \cdot \frac{f}{\frac{\pi}{0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.25:\\ \;\;\;\;\frac{-\log \left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right)}{\pi \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{\frac{\pi}{0}} \cdot \left(-2\right)\\ \end{array} \]

Alternative 7: 97.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.25:\\ \;\;\;\;\frac{-4}{\frac{-\pi}{\log \left(\pi \cdot \left(f \cdot 0.25\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{\frac{\pi}{0}} \cdot \left(-2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.25)
   (/ (- 4.0) (/ (- PI) (log (* PI (* f 0.25)))))
   (* (/ f (/ PI 0.0)) (- 2.0))))

C

double code(double f) {
	double tmp;
	if (f <= 1.25) {
		tmp = -4.0 / (-((double) M_PI) / log((((double) M_PI) * (f * 0.25))));
	} else {
		tmp = (f / (((double) M_PI) / 0.0)) * -2.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.25) {
		tmp = -4.0 / (-Math.PI / Math.log((Math.PI * (f * 0.25))));
	} else {
		tmp = (f / (Math.PI / 0.0)) * -2.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.25:
		tmp = -4.0 / (-math.pi / math.log((math.pi * (f * 0.25))))
	else:
		tmp = (f / (math.pi / 0.0)) * -2.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.25)
		tmp = Float64(Float64(-4.0) / Float64(Float64(-pi) / log(Float64(pi * Float64(f * 0.25)))));
	else
		tmp = Float64(Float64(f / Float64(pi / 0.0)) * Float64(-2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.25)
		tmp = -4.0 / (-pi / log((pi * (f * 0.25))));
	else
		tmp = (f / (pi / 0.0)) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.25], N[((-4.0) / N[((-Pi) / N[Log[N[(Pi * N[(f * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(f / N[(Pi / 0.0), $MachinePrecision]), $MachinePrecision] * (-2.0)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < 1.25

   1. Initial program 6.1%

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

   2. Taylor expanded in f around 0 98.4%

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

      1. associate-*r/ 98.4%

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

      2. associate-/l* 98.3%

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

      3. mul-1-neg 98.3%

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

      4. unsub-neg 98.3%

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

      5. distribute-rgt-out-- 98.3%

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

      6. metadata-eval 98.3%

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

   4. Simplified 98.3%

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

      1. add-exp-log 97.2%

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

      2. diff-log 97.2%

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

   6. Applied egg-rr 97.2%

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

      1. rem-exp-log 98.4%

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

      2. div-inv 98.2%

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

      3. frac-2neg 98.2%

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

      4. metadata-eval 98.2%

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

      5. neg-log 98.2%

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

      6. clear-num 98.2%

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

      7. *-un-lft-identity 98.2%

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

      8. associate-/r* 98.2%

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

      9. div-inv 98.2%

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

      10. metadata-eval 98.2%

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

      11. times-frac 98.2%

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

      12. clear-num 98.2%

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

      13. div-inv 98.2%

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

      14. metadata-eval 98.2%

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

   8. Applied egg-rr 98.2%

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

      1. *-commutative 98.2%

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

      2. associate-*l/ 98.4%

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

      3. neg-mul-1 98.4%

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

      4. associate-*r/ 98.4%

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

      5. *-commutative 98.4%

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

      6. associate-*r* 98.4%

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

      7. *-commutative 98.4%

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

      8. associate-/l* 98.4%

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

      9. metadata-eval 98.4%

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

      10. associate-/l* 98.4%

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

      11. *-lft-identity 98.4%

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

      12. associate-/l* 98.4%

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

      13. associate-/r* 98.4%

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

      14. *-commutative 98.4%

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

      15. associate-/r/ 98.4%

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

      16. metadata-eval 98.4%

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

      17. *-commutative 98.4%

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

      18. associate-*r* 98.4%

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

      19. *-commutative 98.4%

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

      20. *-commutative 98.4%

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

   10. Simplified 98.4%

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

   if 1.25 < f

   1. Initial program 0.9%

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

   3. Applied egg-rr 2.8%

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

   4. Taylor expanded in f around 0 75.9%

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

      1. associate-/l* 75.9%

         \[\leadsto -2 \cdot \color{blue}{\frac{f}{\frac{\pi}{-0.25 \cdot \pi + 0.25 \cdot \pi}}} \]

      2. distribute-rgt-out 75.9%

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

      3. metadata-eval 75.9%

         \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\pi \cdot \color{blue}{0}}} \]

      4. mul0-rgt 75.9%

         \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\color{blue}{0}}} \]

   6. Simplified 75.9%

      \[\leadsto -\color{blue}{2 \cdot \frac{f}{\frac{\pi}{0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.25:\\ \;\;\;\;\frac{-4}{\frac{-\pi}{\log \left(\pi \cdot \left(f \cdot 0.25\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{\frac{\pi}{0}} \cdot \left(-2\right)\\ \end{array} \]

Alternative 8: 97.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.25:\\ \;\;\;\;\log \left(\frac{4}{\pi \cdot f}\right) \cdot \frac{-4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{\frac{\pi}{0}} \cdot \left(-2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.25)
   (* (log (/ 4.0 (* PI f))) (/ (- 4.0) PI))
   (* (/ f (/ PI 0.0)) (- 2.0))))

C

double code(double f) {
	double tmp;
	if (f <= 1.25) {
		tmp = log((4.0 / (((double) M_PI) * f))) * (-4.0 / ((double) M_PI));
	} else {
		tmp = (f / (((double) M_PI) / 0.0)) * -2.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.25) {
		tmp = Math.log((4.0 / (Math.PI * f))) * (-4.0 / Math.PI);
	} else {
		tmp = (f / (Math.PI / 0.0)) * -2.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.25:
		tmp = math.log((4.0 / (math.pi * f))) * (-4.0 / math.pi)
	else:
		tmp = (f / (math.pi / 0.0)) * -2.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.25)
		tmp = Float64(log(Float64(4.0 / Float64(pi * f))) * Float64(Float64(-4.0) / pi));
	else
		tmp = Float64(Float64(f / Float64(pi / 0.0)) * Float64(-2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.25)
		tmp = log((4.0 / (pi * f))) * (-4.0 / pi);
	else
		tmp = (f / (pi / 0.0)) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.25], N[(N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-4.0) / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(f / N[(Pi / 0.0), $MachinePrecision]), $MachinePrecision] * (-2.0)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < 1.25

   1. Initial program 6.1%

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

   2. Taylor expanded in f around 0 98.4%

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

      1. associate-*r/ 98.4%

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

      2. associate-/l* 98.3%

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

      3. mul-1-neg 98.3%

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

      4. unsub-neg 98.3%

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

      5. distribute-rgt-out-- 98.3%

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

      6. metadata-eval 98.3%

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

   4. Simplified 98.3%

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

      1. associate-/r/ 98.3%

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

      2. diff-log 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in f around 0 98.2%

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

      1. *-commutative 98.2%

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

   9. Simplified 98.2%

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

   if 1.25 < f

   1. Initial program 0.9%

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

   3. Applied egg-rr 2.8%

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

   4. Taylor expanded in f around 0 75.9%

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

      1. associate-/l* 75.9%

         \[\leadsto -2 \cdot \color{blue}{\frac{f}{\frac{\pi}{-0.25 \cdot \pi + 0.25 \cdot \pi}}} \]

      2. distribute-rgt-out 75.9%

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

      3. metadata-eval 75.9%

         \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\pi \cdot \color{blue}{0}}} \]

      4. mul0-rgt 75.9%

         \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\color{blue}{0}}} \]

   6. Simplified 75.9%

      \[\leadsto -\color{blue}{2 \cdot \frac{f}{\frac{\pi}{0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.25:\\ \;\;\;\;\log \left(\frac{4}{\pi \cdot f}\right) \cdot \frac{-4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{\frac{\pi}{0}} \cdot \left(-2\right)\\ \end{array} \]

Alternative 9: 5.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \frac{f}{\frac{\pi}{0}} \cdot \left(-2\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ f (/ PI 0.0)) (- 2.0)))

C

double code(double f) {
	return (f / (((double) M_PI) / 0.0)) * -2.0;
}

Java

public static double code(double f) {
	return (f / (Math.PI / 0.0)) * -2.0;
}

Python

def code(f):
	return (f / (math.pi / 0.0)) * -2.0

Julia

function code(f)
	return Float64(Float64(f / Float64(pi / 0.0)) * Float64(-2.0))
end

MATLAB

function tmp = code(f)
	tmp = (f / (pi / 0.0)) * -2.0;
end

Wolfram

code[f_] := N[(N[(f / N[(Pi / 0.0), $MachinePrecision]), $MachinePrecision] * (-2.0)), $MachinePrecision]

Derivation

1. Initial program 6.0%

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

3. Applied egg-rr 3.4%

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

4. Taylor expanded in f around 0 5.4%

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

   1. associate-/l* 5.4%

      \[\leadsto -2 \cdot \color{blue}{\frac{f}{\frac{\pi}{-0.25 \cdot \pi + 0.25 \cdot \pi}}} \]

   2. distribute-rgt-out 5.4%

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

   3. metadata-eval 5.4%

      \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\pi \cdot \color{blue}{0}}} \]

   4. mul0-rgt 5.4%

      \[\leadsto -2 \cdot \frac{f}{\frac{\pi}{\color{blue}{0}}} \]

6. Simplified 5.4%

   \[\leadsto -\color{blue}{2 \cdot \frac{f}{\frac{\pi}{0}}} \]

7. Final simplification 5.4%

   \[\leadsto \frac{f}{\frac{\pi}{0}} \cdot \left(-2\right) \]

Reproduce

herbie shell --seed 2023313 
(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)))))))))