VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 97.0%

Time: 37.3s

Alternatives: 13

Speedup: 5.0×

• could not determine a ground truth

  1. f = -5.4283233981704296e+138

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: 97.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{4} \cdot f}\\ t_1 := e^{-0.25 \cdot \left(\pi \cdot f\right)}\\ t_2 := e^{\left(\pi \cdot f\right) \cdot 0.25}\\ t_3 := e^{\frac{\pi}{4} \cdot \left(-f\right)}\\ \mathbf{if}\;\frac{t_0 + t_3}{t_0 - t_3} \leq 50:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{t_1 + t_2}{t_2 - t_1}\right)}{\sqrt[3]{{\pi}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\frac{4}{\pi}\right) + \mathsf{fma}\left(0.5, \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right), -\log f\right)\right) \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (* (/ PI 4.0) f)))
        (t_1 (exp (* -0.25 (* PI f))))
        (t_2 (exp (* (* PI f) 0.25)))
        (t_3 (exp (* (/ PI 4.0) (- f)))))
   (if (<= (/ (+ t_0 t_3) (- t_0 t_3)) 50.0)
     (* -4.0 (/ (log (/ (+ t_1 t_2) (- t_2 t_1))) (cbrt (pow PI 3.0))))
     (*
      (+
       (log (/ 4.0 PI))
       (fma
        0.5
        (fma
         f
         0.0
         (* (pow f 2.0) (fma 0.5 (* PI (* PI 0.08333333333333333)) 0.0)))
        (- (log f))))
      (/ -1.0 (/ PI 4.0))))))

C

double code(double f) {
	double t_0 = exp(((((double) M_PI) / 4.0) * f));
	double t_1 = exp((-0.25 * (((double) M_PI) * f)));
	double t_2 = exp(((((double) M_PI) * f) * 0.25));
	double t_3 = exp(((((double) M_PI) / 4.0) * -f));
	double tmp;
	if (((t_0 + t_3) / (t_0 - t_3)) <= 50.0) {
		tmp = -4.0 * (log(((t_1 + t_2) / (t_2 - t_1))) / cbrt(pow(((double) M_PI), 3.0)));
	} else {
		tmp = (log((4.0 / ((double) M_PI))) + fma(0.5, fma(f, 0.0, (pow(f, 2.0) * fma(0.5, (((double) M_PI) * (((double) M_PI) * 0.08333333333333333)), 0.0))), -log(f))) * (-1.0 / (((double) M_PI) / 4.0));
	}
	return tmp;
}

Julia

function code(f)
	t_0 = exp(Float64(Float64(pi / 4.0) * f))
	t_1 = exp(Float64(-0.25 * Float64(pi * f)))
	t_2 = exp(Float64(Float64(pi * f) * 0.25))
	t_3 = exp(Float64(Float64(pi / 4.0) * Float64(-f)))
	tmp = 0.0
	if (Float64(Float64(t_0 + t_3) / Float64(t_0 - t_3)) <= 50.0)
		tmp = Float64(-4.0 * Float64(log(Float64(Float64(t_1 + t_2) / Float64(t_2 - t_1))) / cbrt((pi ^ 3.0))));
	else
		tmp = Float64(Float64(log(Float64(4.0 / pi)) + fma(0.5, fma(f, 0.0, Float64((f ^ 2.0) * fma(0.5, Float64(pi * Float64(pi * 0.08333333333333333)), 0.0))), Float64(-log(f)))) * Float64(-1.0 / Float64(pi / 4.0)));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(-0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(Pi * f), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * (-f)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 + t$95$3), $MachinePrecision] / N[(t$95$0 - t$95$3), $MachinePrecision]), $MachinePrecision], 50.0], N[(-4.0 * N[(N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$2 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(f * 0.0 + N[(N[Power[f, 2.0], $MachinePrecision] * N[(0.5 * N[(Pi * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Log[f], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f))))) < 50

   1. Initial program 75.2%

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

      1. distribute-lft-neg-in 75.2%

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

      2. *-commutative 75.2%

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

      3. associate-/r/ 75.2%

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

      4. associate-*l/ 75.2%

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

      5. metadata-eval 75.2%

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

      6. distribute-neg-frac 75.2%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in f around inf 75.2%

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

   5. Taylor expanded in f around inf 75.3%

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

      1. add-cbrt-cube 75.8%

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

      2. pow3 75.8%

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

   7. Applied egg-rr 75.8%

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

   if 50 < (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))))

   1. Initial program 4.8%

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

   2. Taylor expanded in f around 0 95.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. fma-def 95.4%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(f, 0.25 \cdot \pi - -0.25 \cdot \pi, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]

      2. distribute-rgt-out-- 95.4%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

      3. metadata-eval 95.4%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

      4. associate-+r+ 95.4%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)}\right)}\right) \]

      5. +-commutative 95.4%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)} + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right) \]

   4. Simplified 95.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right)}}\right) \]

   5. Taylor expanded in f around 0 95.9%

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

      1. +-commutative 95.9%

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

      2. distribute-lft-out 95.9%

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

      3. fma-def 95.9%

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

   7. Simplified 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \mathsf{fma}\left(0.5, \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right), -\log f\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{e^{\frac{\pi}{4} \cdot f} - e^{\frac{\pi}{4} \cdot \left(-f\right)}} \leq 50:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(\pi \cdot f\right)} + e^{\left(\pi \cdot f\right) \cdot 0.25}}{e^{\left(\pi \cdot f\right) \cdot 0.25} - e^{-0.25 \cdot \left(\pi \cdot f\right)}}\right)}{\sqrt[3]{{\pi}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\frac{4}{\pi}\right) + \mathsf{fma}\left(0.5, \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right), -\log f\right)\right) \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]

Alternative 2: 97.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{\frac{-4}{f}}}\\ t_1 := e^{\frac{\pi}{4} \cdot f}\\ t_2 := {\left(e^{\frac{\pi}{4}}\right)}^{f}\\ t_3 := e^{\frac{\pi}{4} \cdot \left(-f\right)}\\ \mathbf{if}\;\frac{t_1 + t_3}{t_1 - t_3} \leq 50:\\ \;\;\;\;\log \left(\frac{t_0 + t_2}{t_2 - t_0}\right) \cdot \frac{-4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\frac{4}{\pi}\right) + \mathsf{fma}\left(0.5, \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right), -\log f\right)\right) \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (/ PI (/ -4.0 f))))
        (t_1 (exp (* (/ PI 4.0) f)))
        (t_2 (pow (exp (/ PI 4.0)) f))
        (t_3 (exp (* (/ PI 4.0) (- f)))))
   (if (<= (/ (+ t_1 t_3) (- t_1 t_3)) 50.0)
     (* (log (/ (+ t_0 t_2) (- t_2 t_0))) (/ -4.0 PI))
     (*
      (+
       (log (/ 4.0 PI))
       (fma
        0.5
        (fma
         f
         0.0
         (* (pow f 2.0) (fma 0.5 (* PI (* PI 0.08333333333333333)) 0.0)))
        (- (log f))))
      (/ -1.0 (/ PI 4.0))))))

C

double code(double f) {
	double t_0 = exp((((double) M_PI) / (-4.0 / f)));
	double t_1 = exp(((((double) M_PI) / 4.0) * f));
	double t_2 = pow(exp((((double) M_PI) / 4.0)), f);
	double t_3 = exp(((((double) M_PI) / 4.0) * -f));
	double tmp;
	if (((t_1 + t_3) / (t_1 - t_3)) <= 50.0) {
		tmp = log(((t_0 + t_2) / (t_2 - t_0))) * (-4.0 / ((double) M_PI));
	} else {
		tmp = (log((4.0 / ((double) M_PI))) + fma(0.5, fma(f, 0.0, (pow(f, 2.0) * fma(0.5, (((double) M_PI) * (((double) M_PI) * 0.08333333333333333)), 0.0))), -log(f))) * (-1.0 / (((double) M_PI) / 4.0));
	}
	return tmp;
}

Julia

function code(f)
	t_0 = exp(Float64(pi / Float64(-4.0 / f)))
	t_1 = exp(Float64(Float64(pi / 4.0) * f))
	t_2 = exp(Float64(pi / 4.0)) ^ f
	t_3 = exp(Float64(Float64(pi / 4.0) * Float64(-f)))
	tmp = 0.0
	if (Float64(Float64(t_1 + t_3) / Float64(t_1 - t_3)) <= 50.0)
		tmp = Float64(log(Float64(Float64(t_0 + t_2) / Float64(t_2 - t_0))) * Float64(-4.0 / pi));
	else
		tmp = Float64(Float64(log(Float64(4.0 / pi)) + fma(0.5, fma(f, 0.0, Float64((f ^ 2.0) * fma(0.5, Float64(pi * Float64(pi * 0.08333333333333333)), 0.0))), Float64(-log(f)))) * Float64(-1.0 / Float64(pi / 4.0)));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[Exp[N[(Pi / N[(-4.0 / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Exp[N[(Pi / 4.0), $MachinePrecision]], $MachinePrecision], f], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * (-f)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 + t$95$3), $MachinePrecision] / N[(t$95$1 - t$95$3), $MachinePrecision]), $MachinePrecision], 50.0], N[(N[Log[N[(N[(t$95$0 + t$95$2), $MachinePrecision] / N[(t$95$2 - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(f * 0.0 + N[(N[Power[f, 2.0], $MachinePrecision] * N[(0.5 * N[(Pi * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Log[f], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f))))) < 50

   1. Initial program 75.2%

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

      1. distribute-lft-neg-in 75.2%

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

      2. *-commutative 75.2%

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

      3. associate-/r/ 75.2%

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

      4. associate-*l/ 75.2%

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

      5. metadata-eval 75.2%

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

      6. distribute-neg-frac 75.2%

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

   3. Simplified 75.6%

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

   if 50 < (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))))

   1. Initial program 4.8%

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

   2. Taylor expanded in f around 0 95.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. fma-def 95.4%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(f, 0.25 \cdot \pi - -0.25 \cdot \pi, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]

      2. distribute-rgt-out-- 95.4%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

      3. metadata-eval 95.4%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

      4. associate-+r+ 95.4%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)}\right)}\right) \]

      5. +-commutative 95.4%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)} + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right) \]

   4. Simplified 95.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right)}}\right) \]

   5. Taylor expanded in f around 0 95.9%

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

      1. +-commutative 95.9%

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

      2. distribute-lft-out 95.9%

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

      3. fma-def 95.9%

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

   7. Simplified 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \mathsf{fma}\left(0.5, \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right), -\log f\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{e^{\frac{\pi}{4} \cdot f} - e^{\frac{\pi}{4} \cdot \left(-f\right)}} \leq 50:\\ \;\;\;\;\log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + {\left(e^{\frac{\pi}{4}}\right)}^{f}}{{\left(e^{\frac{\pi}{4}}\right)}^{f} - e^{\frac{\pi}{\frac{-4}{f}}}}\right) \cdot \frac{-4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\frac{4}{\pi}\right) + \mathsf{fma}\left(0.5, \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right), -\log f\right)\right) \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]

Alternative 3: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    (+ (exp (* (/ PI 4.0) f)) (exp (* (/ PI 4.0) (- f))))
    (fma
     f
     (* PI 0.5)
     (fma
      (pow f 5.0)
      (* (pow PI 5.0) 1.6276041666666666e-5)
      (fma
       (pow f 3.0)
       (* (pow PI 3.0) 0.005208333333333333)
       (* (pow f 7.0) (* (pow PI 7.0) 2.422030009920635e-8)))))))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(((exp(((((double) M_PI) / 4.0) * f)) + exp(((((double) M_PI) / 4.0) * -f))) / fma(f, (((double) M_PI) * 0.5), fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), (pow(f, 7.0) * (pow(((double) M_PI), 7.0) * 2.422030009920635e-8))))))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(Float64(Float64(exp(Float64(Float64(pi / 4.0) * f)) + exp(Float64(Float64(pi / 4.0) * Float64(-f)))) / fma(f, Float64(pi * 0.5), fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), Float64((f ^ 7.0) * Float64((pi ^ 7.0) * 2.422030009920635e-8))))))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * (-f)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(f * N[(Pi * 0.5), $MachinePrecision] + N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision] + N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(N[Power[f, 7.0], $MachinePrecision] * N[(N[Power[Pi, 7.0], $MachinePrecision] * 2.422030009920635e-8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

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

2. Taylor expanded in f around 0 93.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]
3. Step-by-step derivation

   1. fma-def 93.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(f, 0.25 \cdot \pi - -0.25 \cdot \pi, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]

   2. distribute-rgt-out-- 93.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   3. metadata-eval 93.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   4. associate-+r+ 93.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)}\right)}\right) \]

   5. +-commutative 93.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)} + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right) \]

4. Simplified 93.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right)}}\right) \]

5. Final simplification 93.9%

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}} \]

Alternative 4: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{4} \cdot f}\\ t_1 := e^{-0.25 \cdot \left(\pi \cdot f\right)}\\ t_2 := e^{\left(\pi \cdot f\right) \cdot 0.25}\\ t_3 := e^{\frac{\pi}{4} \cdot \left(-f\right)}\\ \mathbf{if}\;\frac{t_0 + t_3}{t_0 - t_3} \leq 50:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{t_1 + t_2}{t_2 - t_1}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\frac{4}{\pi}\right) + \mathsf{fma}\left(0.5, \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right), -\log f\right)\right) \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (* (/ PI 4.0) f)))
        (t_1 (exp (* -0.25 (* PI f))))
        (t_2 (exp (* (* PI f) 0.25)))
        (t_3 (exp (* (/ PI 4.0) (- f)))))
   (if (<= (/ (+ t_0 t_3) (- t_0 t_3)) 50.0)
     (* -4.0 (/ (log (/ (+ t_1 t_2) (- t_2 t_1))) PI))
     (*
      (+
       (log (/ 4.0 PI))
       (fma
        0.5
        (fma
         f
         0.0
         (* (pow f 2.0) (fma 0.5 (* PI (* PI 0.08333333333333333)) 0.0)))
        (- (log f))))
      (/ -1.0 (/ PI 4.0))))))

C

double code(double f) {
	double t_0 = exp(((((double) M_PI) / 4.0) * f));
	double t_1 = exp((-0.25 * (((double) M_PI) * f)));
	double t_2 = exp(((((double) M_PI) * f) * 0.25));
	double t_3 = exp(((((double) M_PI) / 4.0) * -f));
	double tmp;
	if (((t_0 + t_3) / (t_0 - t_3)) <= 50.0) {
		tmp = -4.0 * (log(((t_1 + t_2) / (t_2 - t_1))) / ((double) M_PI));
	} else {
		tmp = (log((4.0 / ((double) M_PI))) + fma(0.5, fma(f, 0.0, (pow(f, 2.0) * fma(0.5, (((double) M_PI) * (((double) M_PI) * 0.08333333333333333)), 0.0))), -log(f))) * (-1.0 / (((double) M_PI) / 4.0));
	}
	return tmp;
}

Julia

function code(f)
	t_0 = exp(Float64(Float64(pi / 4.0) * f))
	t_1 = exp(Float64(-0.25 * Float64(pi * f)))
	t_2 = exp(Float64(Float64(pi * f) * 0.25))
	t_3 = exp(Float64(Float64(pi / 4.0) * Float64(-f)))
	tmp = 0.0
	if (Float64(Float64(t_0 + t_3) / Float64(t_0 - t_3)) <= 50.0)
		tmp = Float64(-4.0 * Float64(log(Float64(Float64(t_1 + t_2) / Float64(t_2 - t_1))) / pi));
	else
		tmp = Float64(Float64(log(Float64(4.0 / pi)) + fma(0.5, fma(f, 0.0, Float64((f ^ 2.0) * fma(0.5, Float64(pi * Float64(pi * 0.08333333333333333)), 0.0))), Float64(-log(f)))) * Float64(-1.0 / Float64(pi / 4.0)));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(-0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(Pi * f), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * (-f)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 + t$95$3), $MachinePrecision] / N[(t$95$0 - t$95$3), $MachinePrecision]), $MachinePrecision], 50.0], N[(-4.0 * N[(N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$2 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(f * 0.0 + N[(N[Power[f, 2.0], $MachinePrecision] * N[(0.5 * N[(Pi * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Log[f], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f))))) < 50

   1. Initial program 75.2%

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

      1. distribute-lft-neg-in 75.2%

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

      2. *-commutative 75.2%

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

      3. associate-/r/ 75.2%

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

      4. associate-*l/ 75.2%

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

      5. metadata-eval 75.2%

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

      6. distribute-neg-frac 75.2%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in f around inf 75.2%

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

   5. Taylor expanded in f around inf 75.3%

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

   if 50 < (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))))

   1. Initial program 4.8%

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

   2. Taylor expanded in f around 0 95.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. fma-def 95.4%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(f, 0.25 \cdot \pi - -0.25 \cdot \pi, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]

      2. distribute-rgt-out-- 95.4%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

      3. metadata-eval 95.4%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

      4. associate-+r+ 95.4%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)}\right)}\right) \]

      5. +-commutative 95.4%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)} + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right) \]

   4. Simplified 95.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right)}}\right) \]

   5. Taylor expanded in f around 0 95.9%

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

      1. +-commutative 95.9%

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

      2. distribute-lft-out 95.9%

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

      3. fma-def 95.9%

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

   7. Simplified 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \mathsf{fma}\left(0.5, \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right), -\log f\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{e^{\frac{\pi}{4} \cdot f} - e^{\frac{\pi}{4} \cdot \left(-f\right)}} \leq 50:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(\pi \cdot f\right)} + e^{\left(\pi \cdot f\right) \cdot 0.25}}{e^{\left(\pi \cdot f\right) \cdot 0.25} - e^{-0.25 \cdot \left(\pi \cdot f\right)}}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\frac{4}{\pi}\right) + \mathsf{fma}\left(0.5, \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right), -\log f\right)\right) \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]

Alternative 5: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := \frac{-1}{\frac{\pi}{4}}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\left(\log \left(\frac{4}{\pi}\right) + \mathsf{fma}\left(0.5, \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right), -\log f\right)\right) \cdot t_1\\ \mathbf{elif}\;t_0 \leq 200:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + 1}{0.5}\right) \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (/ -1.0 (/ PI 4.0))))
   (if (<= t_0 5e-6)
     (*
      (+
       (log (/ 4.0 PI))
       (fma
        0.5
        (fma
         f
         0.0
         (* (pow f 2.0) (fma 0.5 (* PI (* PI 0.08333333333333333)) 0.0)))
        (- (log f))))
      t_1)
     (if (<= t_0 200.0)
       (*
        (/ 1.0 (/ PI 4.0))
        (log
         (/
          (- (pow (exp 0.25) (* PI f)) (pow (exp -0.25) (* PI f)))
          (* 2.0 (cosh (/ (* PI f) 4.0))))))
       (* (log (/ (+ (exp (* (/ PI 4.0) (- f))) 1.0) 0.5)) t_1)))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = -1.0 / (((double) M_PI) / 4.0);
	double tmp;
	if (t_0 <= 5e-6) {
		tmp = (log((4.0 / ((double) M_PI))) + fma(0.5, fma(f, 0.0, (pow(f, 2.0) * fma(0.5, (((double) M_PI) * (((double) M_PI) * 0.08333333333333333)), 0.0))), -log(f))) * t_1;
	} else if (t_0 <= 200.0) {
		tmp = (1.0 / (((double) M_PI) / 4.0)) * log(((pow(exp(0.25), (((double) M_PI) * f)) - pow(exp(-0.25), (((double) M_PI) * f))) / (2.0 * cosh(((((double) M_PI) * f) / 4.0)))));
	} else {
		tmp = log(((exp(((((double) M_PI) / 4.0) * -f)) + 1.0) / 0.5)) * t_1;
	}
	return tmp;
}

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = Float64(-1.0 / Float64(pi / 4.0))
	tmp = 0.0
	if (t_0 <= 5e-6)
		tmp = Float64(Float64(log(Float64(4.0 / pi)) + fma(0.5, fma(f, 0.0, Float64((f ^ 2.0) * fma(0.5, Float64(pi * Float64(pi * 0.08333333333333333)), 0.0))), Float64(-log(f)))) * t_1);
	elseif (t_0 <= 200.0)
		tmp = Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64((exp(0.25) ^ Float64(pi * f)) - (exp(-0.25) ^ Float64(pi * f))) / Float64(2.0 * cosh(Float64(Float64(pi * f) / 4.0))))));
	else
		tmp = Float64(log(Float64(Float64(exp(Float64(Float64(pi / 4.0) * Float64(-f))) + 1.0) / 0.5)) * t_1);
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-6], N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(f * 0.0 + N[(N[Power[f, 2.0], $MachinePrecision] * N[(0.5 * N[(Pi * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Log[f], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 200.0], N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(N[Power[N[Exp[0.25], $MachinePrecision], N[(Pi * f), $MachinePrecision]], $MachinePrecision] - N[Power[N[Exp[-0.25], $MachinePrecision], N[(Pi * f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Cosh[N[(N[(Pi * f), $MachinePrecision] / 4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(N[(N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * (-f)), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 0.5), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 (PI.f64) 4) f) < 5.00000000000000041e-6

   1. Initial program 5.0%

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

   2. Taylor expanded in f around 0 99.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. fma-def 99.0%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(f, 0.25 \cdot \pi - -0.25 \cdot \pi, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]

      2. distribute-rgt-out-- 99.0%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

      3. metadata-eval 99.0%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

      4. associate-+r+ 99.0%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)}\right)}\right) \]

      5. +-commutative 99.0%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)} + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right) \]

   4. Simplified 99.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right)}}\right) \]

   5. Taylor expanded in f around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-out 99.4%

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

      3. fma-def 99.4%

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

   7. Simplified 99.4%

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

   if 5.00000000000000041e-6 < (*.f64 (/.f64 (PI.f64) 4) f) < 200

   1. Initial program 75.2%

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

   2. Taylor expanded in f around inf 75.2%

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

      1. distribute-lft-neg-in 75.2%

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

      2. metadata-eval 75.2%

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

      3. exp-prod 73.4%

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

      4. exp-prod 73.4%

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

   4. Simplified 73.4%

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

      1. clear-num 73.3%

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

      2. log-div 73.5%

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

      3. metadata-eval 73.5%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\color{blue}{0} - \log \left(\frac{{\left(e^{0.25}\right)}^{\left(f \cdot \pi\right)} - {\left(e^{-0.25}\right)}^{\left(f \cdot \pi\right)}}{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}\right)\right) \]

      4. *-commutative 73.5%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0 - \log \left(\frac{{\left(e^{0.25}\right)}^{\color{blue}{\left(\pi \cdot f\right)}} - {\left(e^{-0.25}\right)}^{\left(f \cdot \pi\right)}}{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}\right)\right) \]

      5. *-commutative 73.5%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0 - \log \left(\frac{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\color{blue}{\left(\pi \cdot f\right)}}}{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}\right)\right) \]

      6. cosh-undef 73.6%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0 - \log \left(\frac{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\color{blue}{2 \cdot \cosh \left(\frac{\pi}{4} \cdot f\right)}}\right)\right) \]

   6. Applied egg-rr 73.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(0 - \log \left(\frac{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}\right)\right)} \]
   7. Step-by-step derivation

      1. neg-sub0 73.6%

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

   8. Simplified 73.6%

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

   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%

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

   4. Taylor expanded in f around 0 3.1%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\color{blue}{1} + e^{-\frac{\pi}{4} \cdot f}}{0.5}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\left(\log \left(\frac{4}{\pi}\right) + \mathsf{fma}\left(0.5, \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right), -\log f\right)\right) \cdot \frac{-1}{\frac{\pi}{4}}\\ \mathbf{elif}\;\frac{\pi}{4} \cdot f \leq 200:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + 1}{0.5}\right) \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]

Alternative 6: 96.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= (* (/ PI 4.0) f) 2e-7)
   (* -4.0 (/ (- (log (/ 4.0 PI)) (log f)) PI))
   (*
    (/ 1.0 (/ PI 4.0))
    (log
     (/
      (- (pow (exp 0.25) (* PI f)) (pow (exp -0.25) (* PI f)))
      (* 2.0 (cosh (/ (* PI f) 4.0))))))))

C

double code(double f) {
	double tmp;
	if (((((double) M_PI) / 4.0) * f) <= 2e-7) {
		tmp = -4.0 * ((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI));
	} else {
		tmp = (1.0 / (((double) M_PI) / 4.0)) * log(((pow(exp(0.25), (((double) M_PI) * f)) - pow(exp(-0.25), (((double) M_PI) * f))) / (2.0 * cosh(((((double) M_PI) * f) / 4.0)))));
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (((Math.PI / 4.0) * f) <= 2e-7) {
		tmp = -4.0 * ((Math.log((4.0 / Math.PI)) - Math.log(f)) / Math.PI);
	} else {
		tmp = (1.0 / (Math.PI / 4.0)) * Math.log(((Math.pow(Math.exp(0.25), (Math.PI * f)) - Math.pow(Math.exp(-0.25), (Math.PI * f))) / (2.0 * Math.cosh(((Math.PI * f) / 4.0)))));
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if ((math.pi / 4.0) * f) <= 2e-7:
		tmp = -4.0 * ((math.log((4.0 / math.pi)) - math.log(f)) / math.pi)
	else:
		tmp = (1.0 / (math.pi / 4.0)) * math.log(((math.pow(math.exp(0.25), (math.pi * f)) - math.pow(math.exp(-0.25), (math.pi * f))) / (2.0 * math.cosh(((math.pi * f) / 4.0)))))
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (Float64(Float64(pi / 4.0) * f) <= 2e-7)
		tmp = Float64(-4.0 * Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi));
	else
		tmp = Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64((exp(0.25) ^ Float64(pi * f)) - (exp(-0.25) ^ Float64(pi * f))) / Float64(2.0 * cosh(Float64(Float64(pi * f) / 4.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (((pi / 4.0) * f) <= 2e-7)
		tmp = -4.0 * ((log((4.0 / pi)) - log(f)) / pi);
	else
		tmp = (1.0 / (pi / 4.0)) * log((((exp(0.25) ^ (pi * f)) - (exp(-0.25) ^ (pi * f))) / (2.0 * cosh(((pi * f) / 4.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision], 2e-7], N[(-4.0 * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(N[Power[N[Exp[0.25], $MachinePrecision], N[(Pi * f), $MachinePrecision]], $MachinePrecision] - N[Power[N[Exp[-0.25], $MachinePrecision], N[(Pi * f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Cosh[N[(N[(Pi * f), $MachinePrecision] / 4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) 4) f) < 1.9999999999999999e-7

   1. Initial program 4.6%

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

      1. distribute-lft-neg-in 4.6%

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

      2. *-commutative 4.6%

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

      3. associate-/r/ 4.6%

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

      4. associate-*l/ 4.6%

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

      5. metadata-eval 4.6%

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

      6. distribute-neg-frac 4.6%

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

   3. Simplified 4.6%

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

   4. Taylor expanded in f around inf 4.6%

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

   5. Taylor expanded in f around inf 4.6%

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

   6. Taylor expanded in f around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. distribute-rgt-out-- 99.5%

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

      3. metadata-eval 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 99.5%

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

      6. metadata-eval 99.5%

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

      7. unsub-neg 99.5%

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

   8. Simplified 99.5%

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

   if 1.9999999999999999e-7 < (*.f64 (/.f64 (PI.f64) 4) f)

   1. Initial program 40.1%

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

   2. Taylor expanded in f around inf 40.1%

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

      1. distribute-lft-neg-in 40.1%

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

      2. metadata-eval 40.1%

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

      3. exp-prod 39.2%

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

      4. exp-prod 39.2%

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

   4. Simplified 39.2%

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

      1. clear-num 39.2%

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

      2. log-div 39.3%

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

      3. metadata-eval 39.3%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\color{blue}{0} - \log \left(\frac{{\left(e^{0.25}\right)}^{\left(f \cdot \pi\right)} - {\left(e^{-0.25}\right)}^{\left(f \cdot \pi\right)}}{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}\right)\right) \]

      4. *-commutative 39.3%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0 - \log \left(\frac{{\left(e^{0.25}\right)}^{\color{blue}{\left(\pi \cdot f\right)}} - {\left(e^{-0.25}\right)}^{\left(f \cdot \pi\right)}}{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}\right)\right) \]

      5. *-commutative 39.3%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0 - \log \left(\frac{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\color{blue}{\left(\pi \cdot f\right)}}}{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}\right)\right) \]

      6. cosh-undef 39.3%

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0 - \log \left(\frac{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\color{blue}{2 \cdot \cosh \left(\frac{\pi}{4} \cdot f\right)}}\right)\right) \]

   6. Applied egg-rr 39.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(0 - \log \left(\frac{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}\right)\right)} \]
   7. Step-by-step derivation

      1. neg-sub0 39.3%

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

   8. Simplified 39.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(-\log \left(\frac{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}\right)\\ \end{array} \]

Alternative 7: 96.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(\frac{2}{\frac{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\cosh \left(\frac{\pi}{\frac{4}{f}}\right)}}\right)}{\pi \cdot 0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= (* (/ PI 4.0) f) 2e-7)
   (* -4.0 (/ (- (log (/ 4.0 PI)) (log f)) PI))
   (/
    (-
     (log
      (/
       2.0
       (/
        (- (pow (exp 0.25) (* PI f)) (pow (exp -0.25) (* PI f)))
        (cosh (/ PI (/ 4.0 f)))))))
    (* PI 0.25))))

C

double code(double f) {
	double tmp;
	if (((((double) M_PI) / 4.0) * f) <= 2e-7) {
		tmp = -4.0 * ((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI));
	} else {
		tmp = -log((2.0 / ((pow(exp(0.25), (((double) M_PI) * f)) - pow(exp(-0.25), (((double) M_PI) * f))) / cosh((((double) M_PI) / (4.0 / f)))))) / (((double) M_PI) * 0.25);
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (((Math.PI / 4.0) * f) <= 2e-7) {
		tmp = -4.0 * ((Math.log((4.0 / Math.PI)) - Math.log(f)) / Math.PI);
	} else {
		tmp = -Math.log((2.0 / ((Math.pow(Math.exp(0.25), (Math.PI * f)) - Math.pow(Math.exp(-0.25), (Math.PI * f))) / Math.cosh((Math.PI / (4.0 / f)))))) / (Math.PI * 0.25);
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if ((math.pi / 4.0) * f) <= 2e-7:
		tmp = -4.0 * ((math.log((4.0 / math.pi)) - math.log(f)) / math.pi)
	else:
		tmp = -math.log((2.0 / ((math.pow(math.exp(0.25), (math.pi * f)) - math.pow(math.exp(-0.25), (math.pi * f))) / math.cosh((math.pi / (4.0 / f)))))) / (math.pi * 0.25)
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (Float64(Float64(pi / 4.0) * f) <= 2e-7)
		tmp = Float64(-4.0 * Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi));
	else
		tmp = Float64(Float64(-log(Float64(2.0 / Float64(Float64((exp(0.25) ^ Float64(pi * f)) - (exp(-0.25) ^ Float64(pi * f))) / cosh(Float64(pi / Float64(4.0 / f))))))) / Float64(pi * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (((pi / 4.0) * f) <= 2e-7)
		tmp = -4.0 * ((log((4.0 / pi)) - log(f)) / pi);
	else
		tmp = -log((2.0 / (((exp(0.25) ^ (pi * f)) - (exp(-0.25) ^ (pi * f))) / cosh((pi / (4.0 / f)))))) / (pi * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision], 2e-7], N[(-4.0 * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[((-N[Log[N[(2.0 / N[(N[(N[Power[N[Exp[0.25], $MachinePrecision], N[(Pi * f), $MachinePrecision]], $MachinePrecision] - N[Power[N[Exp[-0.25], $MachinePrecision], N[(Pi * f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Cosh[N[(Pi / N[(4.0 / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) 4) f) < 1.9999999999999999e-7

   1. Initial program 4.6%

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

      1. distribute-lft-neg-in 4.6%

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

      2. *-commutative 4.6%

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

      3. associate-/r/ 4.6%

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

      4. associate-*l/ 4.6%

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

      5. metadata-eval 4.6%

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

      6. distribute-neg-frac 4.6%

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

   3. Simplified 4.6%

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

   4. Taylor expanded in f around inf 4.6%

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

   5. Taylor expanded in f around inf 4.6%

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

   6. Taylor expanded in f around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. distribute-rgt-out-- 99.5%

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

      3. metadata-eval 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 99.5%

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

      6. metadata-eval 99.5%

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

      7. unsub-neg 99.5%

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

   8. Simplified 99.5%

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

   if 1.9999999999999999e-7 < (*.f64 (/.f64 (PI.f64) 4) f)

   1. Initial program 40.1%

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

   2. Taylor expanded in f around inf 40.1%

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

      1. distribute-lft-neg-in 40.1%

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

      2. metadata-eval 40.1%

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

      3. exp-prod 39.2%

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

      4. exp-prod 39.2%

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

   4. Simplified 39.2%

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

      1. expm1-log1p-u 39.2%

         \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{{\left(e^{0.25}\right)}^{\left(f \cdot \pi\right)} - {\left(e^{-0.25}\right)}^{\left(f \cdot \pi\right)}}\right)\right)\right)} \]

      2. expm1-udef 39.3%

         \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{{\left(e^{0.25}\right)}^{\left(f \cdot \pi\right)} - {\left(e^{-0.25}\right)}^{\left(f \cdot \pi\right)}}\right)\right)} - 1\right)} \]

   6. Applied egg-rr 39.3%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right)}{\pi \cdot 0.25}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 39.3%

         \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right)}{\pi \cdot 0.25}\right)\right)} \]

      2. expm1-log1p 39.3%

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

      3. associate-/l* 39.3%

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

      4. associate-/l* 39.3%

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

   8. Simplified 39.3%

      \[\leadsto -\color{blue}{\frac{\log \left(\frac{2}{\frac{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\cosh \left(\frac{\pi}{\frac{4}{f}}\right)}}\right)}{\pi \cdot 0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(\frac{2}{\frac{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\cosh \left(\frac{\pi}{\frac{4}{f}}\right)}}\right)}{\pi \cdot 0.25}\\ \end{array} \]

Alternative 8: 96.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right)}{\pi \cdot 0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= (* (/ PI 4.0) f) 2e-7)
   (* -4.0 (/ (- (log (/ 4.0 PI)) (log f)) PI))
   (/
    (-
     (log
      (/
       (* 2.0 (cosh (/ (* PI f) 4.0)))
       (- (pow (exp 0.25) (* PI f)) (pow (exp -0.25) (* PI f))))))
    (* PI 0.25))))

C

double code(double f) {
	double tmp;
	if (((((double) M_PI) / 4.0) * f) <= 2e-7) {
		tmp = -4.0 * ((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI));
	} else {
		tmp = -log(((2.0 * cosh(((((double) M_PI) * f) / 4.0))) / (pow(exp(0.25), (((double) M_PI) * f)) - pow(exp(-0.25), (((double) M_PI) * f))))) / (((double) M_PI) * 0.25);
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (((Math.PI / 4.0) * f) <= 2e-7) {
		tmp = -4.0 * ((Math.log((4.0 / Math.PI)) - Math.log(f)) / Math.PI);
	} else {
		tmp = -Math.log(((2.0 * Math.cosh(((Math.PI * f) / 4.0))) / (Math.pow(Math.exp(0.25), (Math.PI * f)) - Math.pow(Math.exp(-0.25), (Math.PI * f))))) / (Math.PI * 0.25);
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if ((math.pi / 4.0) * f) <= 2e-7:
		tmp = -4.0 * ((math.log((4.0 / math.pi)) - math.log(f)) / math.pi)
	else:
		tmp = -math.log(((2.0 * math.cosh(((math.pi * f) / 4.0))) / (math.pow(math.exp(0.25), (math.pi * f)) - math.pow(math.exp(-0.25), (math.pi * f))))) / (math.pi * 0.25)
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (Float64(Float64(pi / 4.0) * f) <= 2e-7)
		tmp = Float64(-4.0 * Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi));
	else
		tmp = Float64(Float64(-log(Float64(Float64(2.0 * cosh(Float64(Float64(pi * f) / 4.0))) / Float64((exp(0.25) ^ Float64(pi * f)) - (exp(-0.25) ^ Float64(pi * f)))))) / Float64(pi * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (((pi / 4.0) * f) <= 2e-7)
		tmp = -4.0 * ((log((4.0 / pi)) - log(f)) / pi);
	else
		tmp = -log(((2.0 * cosh(((pi * f) / 4.0))) / ((exp(0.25) ^ (pi * f)) - (exp(-0.25) ^ (pi * f))))) / (pi * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision], 2e-7], N[(-4.0 * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[((-N[Log[N[(N[(2.0 * N[Cosh[N[(N[(Pi * f), $MachinePrecision] / 4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Exp[0.25], $MachinePrecision], N[(Pi * f), $MachinePrecision]], $MachinePrecision] - N[Power[N[Exp[-0.25], $MachinePrecision], N[(Pi * f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) 4) f) < 1.9999999999999999e-7

   1. Initial program 4.6%

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

      1. distribute-lft-neg-in 4.6%

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

      2. *-commutative 4.6%

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

      3. associate-/r/ 4.6%

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

      4. associate-*l/ 4.6%

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

      5. metadata-eval 4.6%

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

      6. distribute-neg-frac 4.6%

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

   3. Simplified 4.6%

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

   4. Taylor expanded in f around inf 4.6%

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

   5. Taylor expanded in f around inf 4.6%

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

   6. Taylor expanded in f around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. distribute-rgt-out-- 99.5%

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

      3. metadata-eval 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 99.5%

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

      6. metadata-eval 99.5%

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

      7. unsub-neg 99.5%

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

   8. Simplified 99.5%

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

   if 1.9999999999999999e-7 < (*.f64 (/.f64 (PI.f64) 4) f)

   1. Initial program 40.1%

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

   2. Taylor expanded in f around inf 40.1%

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

      1. distribute-lft-neg-in 40.1%

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

      2. metadata-eval 40.1%

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

      3. exp-prod 39.2%

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

      4. exp-prod 39.2%

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

   4. Simplified 39.2%

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

      1. associate-*l/ 39.3%

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

   6. Applied egg-rr 39.3%

      \[\leadsto -\color{blue}{\frac{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right)}{\pi \cdot 0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{{\left(e^{0.25}\right)}^{\left(\pi \cdot f\right)} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right)}{\pi \cdot 0.25}\\ \end{array} \]

Alternative 9: 95.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.3%

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

   1. distribute-lft-neg-in 7.3%

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

   2. *-commutative 7.3%

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

   3. associate-/r/ 7.3%

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

   4. associate-*l/ 7.3%

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

   5. metadata-eval 7.3%

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

   6. distribute-neg-frac 7.3%

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

3. Simplified 7.3%

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

4. Taylor expanded in f around inf 7.3%

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

5. Taylor expanded in f around inf 7.3%

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

6. Taylor expanded in f around 0 93.2%

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

   1. mul-1-neg 93.2%

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

   2. distribute-rgt-out-- 93.2%

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

   3. metadata-eval 93.2%

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

   4. *-commutative 93.2%

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

   5. associate-/r* 93.2%

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

   6. metadata-eval 93.2%

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

   7. unsub-neg 93.2%

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

8. Simplified 93.2%

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

   1. add-sqr-sqrt 93.2%

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

   2. pow2 93.2%

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

10. Applied egg-rr 93.2%

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

11. Final simplification 93.2%

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

Alternative 10: 95.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.3%

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

   1. distribute-lft-neg-in 7.3%

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

   2. *-commutative 7.3%

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

   3. associate-/r/ 7.3%

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

   4. associate-*l/ 7.3%

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

   5. metadata-eval 7.3%

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

   6. distribute-neg-frac 7.3%

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

3. Simplified 7.3%

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

4. Taylor expanded in f around inf 7.3%

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

5. Taylor expanded in f around inf 7.3%

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

6. Taylor expanded in f around 0 93.2%

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

   1. mul-1-neg 93.2%

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

   2. distribute-rgt-out-- 93.2%

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

   3. metadata-eval 93.2%

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

   4. *-commutative 93.2%

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

   5. associate-/r* 93.2%

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

   6. metadata-eval 93.2%

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

   7. unsub-neg 93.2%

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

8. Simplified 93.2%

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

9. Final simplification 93.2%

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

Alternative 11: 95.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\pi}{4}} \cdot \log \left(f \cdot \frac{\pi \cdot 0.5}{2}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (/ 1.0 (/ PI 4.0)) (log (* f (/ (* PI 0.5) 2.0)))))

C

double code(double f) {
	return (1.0 / (((double) M_PI) / 4.0)) * log((f * ((((double) M_PI) * 0.5) / 2.0)));
}

Java

public static double code(double f) {
	return (1.0 / (Math.PI / 4.0)) * Math.log((f * ((Math.PI * 0.5) / 2.0)));
}

Python

def code(f):
	return (1.0 / (math.pi / 4.0)) * math.log((f * ((math.pi * 0.5) / 2.0)))

Julia

function code(f)
	return Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(f * Float64(Float64(pi * 0.5) / 2.0))))
end

MATLAB

function tmp = code(f)
	tmp = (1.0 / (pi / 4.0)) * log((f * ((pi * 0.5) / 2.0)));
end

Wolfram

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

Derivation

1. Initial program 7.3%

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

2. Taylor expanded in f around 0 92.7%

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

   1. associate-/r* 92.7%

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

   2. distribute-rgt-out-- 92.7%

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

   3. metadata-eval 92.7%

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

4. Simplified 92.7%

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

   1. clear-num 92.7%

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

   2. log-div 92.7%

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

   3. metadata-eval 92.7%

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

6. Applied egg-rr 92.7%

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

   1. neg-sub0 92.7%

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

   2. metadata-eval 92.7%

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

   3. distribute-rgt-out-- 92.7%

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

   4. associate-/r/ 93.1%

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

   5. distribute-rgt-out-- 93.1%

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

   6. metadata-eval 93.1%

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

8. Simplified 93.1%

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

9. Final simplification 93.1%

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

Alternative 12: 95.8% accurate, 3.3× speedup

Math

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

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(-4.0 * Float64(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[(-4.0 * N[(N[Log[N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

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

   1. distribute-lft-neg-in 7.3%

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

   2. *-commutative 7.3%

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

   3. associate-/r/ 7.3%

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

   4. associate-*l/ 7.3%

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

   5. metadata-eval 7.3%

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

   6. distribute-neg-frac 7.3%

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

3. Simplified 7.3%

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

4. Taylor expanded in f around inf 7.3%

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

5. Taylor expanded in f around inf 7.3%

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

6. Taylor expanded in f around 0 93.2%

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

   1. mul-1-neg 93.2%

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

   2. distribute-rgt-out-- 93.2%

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

   3. metadata-eval 93.2%

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

   4. *-commutative 93.2%

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

   5. associate-/r* 93.2%

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

   6. metadata-eval 93.2%

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

   7. unsub-neg 93.2%

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

8. Simplified 93.2%

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

   1. diff-log 92.9%

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

10. Applied egg-rr 92.9%

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

11. Final simplification 92.9%

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

Alternative 13: 14.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.3%

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

3. Applied egg-rr 13.8%

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

   1. clear-num 13.8%

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

   2. log-div 13.8%

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

   3. metadata-eval 13.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\color{blue}{0} - \log \left(\frac{0.5}{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}\right)\right) \]

   4. cosh-undef 13.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0 - \log \left(\frac{0.5}{\color{blue}{2 \cdot \cosh \left(\frac{\pi}{4} \cdot f\right)}}\right)\right) \]

   5. associate-*l/ 13.8%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(0 - \log \left(\frac{0.5}{2 \cdot \cosh \color{blue}{\left(\frac{\pi \cdot f}{4}\right)}}\right)\right) \]

5. Applied egg-rr 13.8%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(0 - \log \left(\frac{0.5}{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}\right)\right)} \]
6. Step-by-step derivation

   1. neg-sub0 13.8%

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

   2. associate-/r* 13.8%

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

   3. metadata-eval 13.8%

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

7. Simplified 13.8%

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

8. Taylor expanded in f around 0 13.9%

   \[\leadsto -\color{blue}{-4 \cdot \frac{\log 0.25}{\pi}} \]

9. Final simplification 13.9%

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

Reproduce

herbie shell --seed 2023314 
(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)))))))))