VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 97.0%

Time: 1.0min

Alternatives: 5

Speedup: 3.3×

• 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{f \cdot \frac{\pi}{4}}\\ t_1 := e^{-0.25 \cdot \left(\pi \cdot f\right)}\\ t_2 := e^{f \cdot \frac{-\pi}{4}}\\ t_3 := e^{0.25 \cdot \left(\pi \cdot f\right)}\\ \mathbf{if}\;\frac{t_0 + t_2}{t_0 - t_2} \leq 400000:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{t_1 + t_3}{t_3 - t_1}\right)}{\sqrt[3]{{\pi}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\pi} \cdot \left(--4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (* f (/ PI 4.0))))
        (t_1 (exp (* -0.25 (* PI f))))
        (t_2 (exp (* f (/ (- PI) 4.0))))
        (t_3 (exp (* 0.25 (* PI f)))))
   (if (<= (/ (+ t_0 t_2) (- t_0 t_2)) 400000.0)
     (* -4.0 (/ (log (/ (+ t_1 t_3) (- t_3 t_1))) (cbrt (pow PI 3.0))))
     (* (/ (log (* f (* PI 0.25))) PI) (- -4.0)))))

C

double code(double f) {
	double t_0 = exp((f * (((double) M_PI) / 4.0)));
	double t_1 = exp((-0.25 * (((double) M_PI) * f)));
	double t_2 = exp((f * (-((double) M_PI) / 4.0)));
	double t_3 = exp((0.25 * (((double) M_PI) * f)));
	double tmp;
	if (((t_0 + t_2) / (t_0 - t_2)) <= 400000.0) {
		tmp = -4.0 * (log(((t_1 + t_3) / (t_3 - t_1))) / cbrt(pow(((double) M_PI), 3.0)));
	} else {
		tmp = (log((f * (((double) M_PI) * 0.25))) / ((double) M_PI)) * -(-4.0);
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = Math.exp((f * (Math.PI / 4.0)));
	double t_1 = Math.exp((-0.25 * (Math.PI * f)));
	double t_2 = Math.exp((f * (-Math.PI / 4.0)));
	double t_3 = Math.exp((0.25 * (Math.PI * f)));
	double tmp;
	if (((t_0 + t_2) / (t_0 - t_2)) <= 400000.0) {
		tmp = -4.0 * (Math.log(((t_1 + t_3) / (t_3 - t_1))) / Math.cbrt(Math.pow(Math.PI, 3.0)));
	} else {
		tmp = (Math.log((f * (Math.PI * 0.25))) / Math.PI) * -(-4.0);
	}
	return tmp;
}

Julia

function code(f)
	t_0 = exp(Float64(f * Float64(pi / 4.0)))
	t_1 = exp(Float64(-0.25 * Float64(pi * f)))
	t_2 = exp(Float64(f * Float64(Float64(-pi) / 4.0)))
	t_3 = exp(Float64(0.25 * Float64(pi * f)))
	tmp = 0.0
	if (Float64(Float64(t_0 + t_2) / Float64(t_0 - t_2)) <= 400000.0)
		tmp = Float64(-4.0 * Float64(log(Float64(Float64(t_1 + t_3) / Float64(t_3 - t_1))) / cbrt((pi ^ 3.0))));
	else
		tmp = Float64(Float64(log(Float64(f * Float64(pi * 0.25))) / pi) * Float64(-(-4.0)));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[Exp[N[(f * N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(-0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(f * N[((-Pi) / 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 + t$95$2), $MachinePrecision] / N[(t$95$0 - t$95$2), $MachinePrecision]), $MachinePrecision], 400000.0], N[(-4.0 * N[(N[Log[N[(N[(t$95$1 + t$95$3), $MachinePrecision] / N[(t$95$3 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[(f * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * (--4.0)), $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))))) < 4e5

   1. Initial program 76.2%

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

      1. distribute-lft-neg-in 76.2%

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

      2. *-commutative 76.2%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in f around inf 76.2%

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

      1. rem-cbrt-cube 76.7%

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

   6. Applied egg-rr 76.7%

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

   if 4e5 < (/.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 4.5%

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

      1. distribute-lft-neg-in 4.5%

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

      2. *-commutative 4.5%

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

   3. Simplified 4.5%

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

   4. Taylor expanded in f around 0 95.4%

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

      1. expm1-log1p-u 0.1%

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

      2. expm1-udef 0.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left(\frac{2}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}\right)} - 1} \]

      3. distribute-rgt-out-- 0.1%

         \[\leadsto e^{\mathsf{log1p}\left(\log \left(\frac{2}{f \cdot \color{blue}{\left(\pi \cdot \left(0.25 - -0.25\right)\right)}}\right) \cdot \frac{-4}{\pi}\right)} - 1 \]

      4. metadata-eval 0.1%

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

   6. Applied egg-rr 0.1%

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

      1. expm1-def 0.1%

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

      2. expm1-log1p 95.4%

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

      3. associate-*r/ 95.6%

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

      4. *-commutative 95.6%

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

      5. *-lft-identity 95.6%

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

      6. times-frac 95.6%

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

      7. metadata-eval 95.6%

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

   8. Simplified 95.6%

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

      1. associate-/l/ 95.6%

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

      2. clear-num 95.6%

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

      3. metadata-eval 95.6%

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

      4. associate-/l* 95.6%

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

      5. associate-*r* 95.6%

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

      6. log-rec 96.0%

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

      7. clear-num 95.6%

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

      8. clear-num 95.6%

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

      9. associate-*r* 95.6%

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

      10. associate-/l* 95.6%

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

      11. metadata-eval 95.6%

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

      12. clear-num 95.6%

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

      13. associate-/l/ 95.6%

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

      14. clear-num 96.0%

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

      15. div-inv 96.0%

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

      16. clear-num 96.0%

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

      17. div-inv 96.0%

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

      18. metadata-eval 96.0%

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

   10. Applied egg-rr 96.0%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{f \cdot \frac{\pi}{4}} + e^{f \cdot \frac{-\pi}{4}}}{e^{f \cdot \frac{\pi}{4}} - e^{f \cdot \frac{-\pi}{4}}} \leq 400000:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(\pi \cdot f\right)} + e^{0.25 \cdot \left(\pi \cdot f\right)}}{e^{0.25 \cdot \left(\pi \cdot f\right)} - e^{-0.25 \cdot \left(\pi \cdot f\right)}}\right)}{\sqrt[3]{{\pi}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\pi} \cdot \left(--4\right)\\ \end{array} \]

Alternative 2: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{f \cdot \frac{\pi}{4}}}{f \cdot \left(\pi \cdot 0.25 - \pi \cdot -0.25\right) + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - {\pi}^{3} \cdot -0.0026041666666666665\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - {\pi}^{5} \cdot -8.138020833333333 \cdot 10^{-6}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - {\pi}^{7} \cdot -1.2110150049603175 \cdot 10^{-8}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    (+ (exp (/ PI (/ -4.0 f))) (exp (* f (/ PI 4.0))))
    (+
     (* f (- (* PI 0.25) (* PI -0.25)))
     (+
      (*
       (pow f 3.0)
       (-
        (* 0.0026041666666666665 (pow PI 3.0))
        (* (pow PI 3.0) -0.0026041666666666665)))
      (+
       (*
        (pow f 5.0)
        (-
         (* 8.138020833333333e-6 (pow PI 5.0))
         (* (pow PI 5.0) -8.138020833333333e-6)))
       (*
        (pow f 7.0)
        (-
         (* 1.2110150049603175e-8 (pow PI 7.0))
         (* (pow PI 7.0) -1.2110150049603175e-8))))))))
  (/ -4.0 PI)))

C

double code(double f) {
	return log(((exp((((double) M_PI) / (-4.0 / f))) + exp((f * (((double) M_PI) / 4.0)))) / ((f * ((((double) M_PI) * 0.25) - (((double) M_PI) * -0.25))) + ((pow(f, 3.0) * ((0.0026041666666666665 * pow(((double) M_PI), 3.0)) - (pow(((double) M_PI), 3.0) * -0.0026041666666666665))) + ((pow(f, 5.0) * ((8.138020833333333e-6 * pow(((double) M_PI), 5.0)) - (pow(((double) M_PI), 5.0) * -8.138020833333333e-6))) + (pow(f, 7.0) * ((1.2110150049603175e-8 * pow(((double) M_PI), 7.0)) - (pow(((double) M_PI), 7.0) * -1.2110150049603175e-8)))))))) * (-4.0 / ((double) M_PI));
}

Java

public static double code(double f) {
	return Math.log(((Math.exp((Math.PI / (-4.0 / f))) + Math.exp((f * (Math.PI / 4.0)))) / ((f * ((Math.PI * 0.25) - (Math.PI * -0.25))) + ((Math.pow(f, 3.0) * ((0.0026041666666666665 * Math.pow(Math.PI, 3.0)) - (Math.pow(Math.PI, 3.0) * -0.0026041666666666665))) + ((Math.pow(f, 5.0) * ((8.138020833333333e-6 * Math.pow(Math.PI, 5.0)) - (Math.pow(Math.PI, 5.0) * -8.138020833333333e-6))) + (Math.pow(f, 7.0) * ((1.2110150049603175e-8 * Math.pow(Math.PI, 7.0)) - (Math.pow(Math.PI, 7.0) * -1.2110150049603175e-8)))))))) * (-4.0 / Math.PI);
}

Python

def code(f):
	return math.log(((math.exp((math.pi / (-4.0 / f))) + math.exp((f * (math.pi / 4.0)))) / ((f * ((math.pi * 0.25) - (math.pi * -0.25))) + ((math.pow(f, 3.0) * ((0.0026041666666666665 * math.pow(math.pi, 3.0)) - (math.pow(math.pi, 3.0) * -0.0026041666666666665))) + ((math.pow(f, 5.0) * ((8.138020833333333e-6 * math.pow(math.pi, 5.0)) - (math.pow(math.pi, 5.0) * -8.138020833333333e-6))) + (math.pow(f, 7.0) * ((1.2110150049603175e-8 * math.pow(math.pi, 7.0)) - (math.pow(math.pi, 7.0) * -1.2110150049603175e-8)))))))) * (-4.0 / math.pi)

Julia

function code(f)
	return Float64(log(Float64(Float64(exp(Float64(pi / Float64(-4.0 / f))) + exp(Float64(f * Float64(pi / 4.0)))) / Float64(Float64(f * Float64(Float64(pi * 0.25) - Float64(pi * -0.25))) + Float64(Float64((f ^ 3.0) * Float64(Float64(0.0026041666666666665 * (pi ^ 3.0)) - Float64((pi ^ 3.0) * -0.0026041666666666665))) + Float64(Float64((f ^ 5.0) * Float64(Float64(8.138020833333333e-6 * (pi ^ 5.0)) - Float64((pi ^ 5.0) * -8.138020833333333e-6))) + Float64((f ^ 7.0) * Float64(Float64(1.2110150049603175e-8 * (pi ^ 7.0)) - Float64((pi ^ 7.0) * -1.2110150049603175e-8)))))))) * Float64(-4.0 / pi))
end

MATLAB

function tmp = code(f)
	tmp = log(((exp((pi / (-4.0 / f))) + exp((f * (pi / 4.0)))) / ((f * ((pi * 0.25) - (pi * -0.25))) + (((f ^ 3.0) * ((0.0026041666666666665 * (pi ^ 3.0)) - ((pi ^ 3.0) * -0.0026041666666666665))) + (((f ^ 5.0) * ((8.138020833333333e-6 * (pi ^ 5.0)) - ((pi ^ 5.0) * -8.138020833333333e-6))) + ((f ^ 7.0) * ((1.2110150049603175e-8 * (pi ^ 7.0)) - ((pi ^ 7.0) * -1.2110150049603175e-8)))))))) * (-4.0 / pi);
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[Exp[N[(Pi / N[(-4.0 / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(f * N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(f * N[(N[(Pi * 0.25), $MachinePrecision] - N[(Pi * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[(0.0026041666666666665 * N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[Pi, 3.0], $MachinePrecision] * -0.0026041666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[(8.138020833333333e-6 * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[Pi, 5.0], $MachinePrecision] * -8.138020833333333e-6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[f, 7.0], $MachinePrecision] * N[(N[(1.2110150049603175e-8 * N[Power[Pi, 7.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[Pi, 7.0], $MachinePrecision] * -1.2110150049603175e-8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / 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. Simplified 7.3%

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

4. Taylor expanded in f around 0 93.9%

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

5. Final simplification 93.9%

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

Alternative 3: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{f \cdot \frac{\pi}{4}}\\ t_1 := e^{-0.25 \cdot \left(\pi \cdot f\right)}\\ t_2 := e^{f \cdot \frac{-\pi}{4}}\\ t_3 := e^{0.25 \cdot \left(\pi \cdot f\right)}\\ \mathbf{if}\;\frac{t_0 + t_2}{t_0 - t_2} \leq 400000:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{t_1 + t_3}{t_3 - t_1}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\pi} \cdot \left(--4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (* f (/ PI 4.0))))
        (t_1 (exp (* -0.25 (* PI f))))
        (t_2 (exp (* f (/ (- PI) 4.0))))
        (t_3 (exp (* 0.25 (* PI f)))))
   (if (<= (/ (+ t_0 t_2) (- t_0 t_2)) 400000.0)
     (* -4.0 (/ (log (/ (+ t_1 t_3) (- t_3 t_1))) PI))
     (* (/ (log (* f (* PI 0.25))) PI) (- -4.0)))))

C

double code(double f) {
	double t_0 = exp((f * (((double) M_PI) / 4.0)));
	double t_1 = exp((-0.25 * (((double) M_PI) * f)));
	double t_2 = exp((f * (-((double) M_PI) / 4.0)));
	double t_3 = exp((0.25 * (((double) M_PI) * f)));
	double tmp;
	if (((t_0 + t_2) / (t_0 - t_2)) <= 400000.0) {
		tmp = -4.0 * (log(((t_1 + t_3) / (t_3 - t_1))) / ((double) M_PI));
	} else {
		tmp = (log((f * (((double) M_PI) * 0.25))) / ((double) M_PI)) * -(-4.0);
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = Math.exp((f * (Math.PI / 4.0)));
	double t_1 = Math.exp((-0.25 * (Math.PI * f)));
	double t_2 = Math.exp((f * (-Math.PI / 4.0)));
	double t_3 = Math.exp((0.25 * (Math.PI * f)));
	double tmp;
	if (((t_0 + t_2) / (t_0 - t_2)) <= 400000.0) {
		tmp = -4.0 * (Math.log(((t_1 + t_3) / (t_3 - t_1))) / Math.PI);
	} else {
		tmp = (Math.log((f * (Math.PI * 0.25))) / Math.PI) * -(-4.0);
	}
	return tmp;
}

Python

def code(f):
	t_0 = math.exp((f * (math.pi / 4.0)))
	t_1 = math.exp((-0.25 * (math.pi * f)))
	t_2 = math.exp((f * (-math.pi / 4.0)))
	t_3 = math.exp((0.25 * (math.pi * f)))
	tmp = 0
	if ((t_0 + t_2) / (t_0 - t_2)) <= 400000.0:
		tmp = -4.0 * (math.log(((t_1 + t_3) / (t_3 - t_1))) / math.pi)
	else:
		tmp = (math.log((f * (math.pi * 0.25))) / math.pi) * -(-4.0)
	return tmp

Julia

function code(f)
	t_0 = exp(Float64(f * Float64(pi / 4.0)))
	t_1 = exp(Float64(-0.25 * Float64(pi * f)))
	t_2 = exp(Float64(f * Float64(Float64(-pi) / 4.0)))
	t_3 = exp(Float64(0.25 * Float64(pi * f)))
	tmp = 0.0
	if (Float64(Float64(t_0 + t_2) / Float64(t_0 - t_2)) <= 400000.0)
		tmp = Float64(-4.0 * Float64(log(Float64(Float64(t_1 + t_3) / Float64(t_3 - t_1))) / pi));
	else
		tmp = Float64(Float64(log(Float64(f * Float64(pi * 0.25))) / pi) * Float64(-(-4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	t_0 = exp((f * (pi / 4.0)));
	t_1 = exp((-0.25 * (pi * f)));
	t_2 = exp((f * (-pi / 4.0)));
	t_3 = exp((0.25 * (pi * f)));
	tmp = 0.0;
	if (((t_0 + t_2) / (t_0 - t_2)) <= 400000.0)
		tmp = -4.0 * (log(((t_1 + t_3) / (t_3 - t_1))) / pi);
	else
		tmp = (log((f * (pi * 0.25))) / pi) * -(-4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := Block[{t$95$0 = N[Exp[N[(f * N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(-0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(f * N[((-Pi) / 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 + t$95$2), $MachinePrecision] / N[(t$95$0 - t$95$2), $MachinePrecision]), $MachinePrecision], 400000.0], N[(-4.0 * N[(N[Log[N[(N[(t$95$1 + t$95$3), $MachinePrecision] / N[(t$95$3 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[(f * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * (--4.0)), $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))))) < 4e5

   1. Initial program 76.2%

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

      1. distribute-lft-neg-in 76.2%

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

      2. *-commutative 76.2%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in f around inf 76.2%

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

   if 4e5 < (/.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 4.5%

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

      1. distribute-lft-neg-in 4.5%

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

      2. *-commutative 4.5%

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

   3. Simplified 4.5%

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

   4. Taylor expanded in f around 0 95.4%

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

      1. expm1-log1p-u 0.1%

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

      2. expm1-udef 0.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left(\frac{2}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}\right)} - 1} \]

      3. distribute-rgt-out-- 0.1%

         \[\leadsto e^{\mathsf{log1p}\left(\log \left(\frac{2}{f \cdot \color{blue}{\left(\pi \cdot \left(0.25 - -0.25\right)\right)}}\right) \cdot \frac{-4}{\pi}\right)} - 1 \]

      4. metadata-eval 0.1%

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

   6. Applied egg-rr 0.1%

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

      1. expm1-def 0.1%

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

      2. expm1-log1p 95.4%

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

      3. associate-*r/ 95.6%

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

      4. *-commutative 95.6%

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

      5. *-lft-identity 95.6%

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

      6. times-frac 95.6%

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

      7. metadata-eval 95.6%

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

   8. Simplified 95.6%

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

      1. associate-/l/ 95.6%

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

      2. clear-num 95.6%

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

      3. metadata-eval 95.6%

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

      4. associate-/l* 95.6%

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

      5. associate-*r* 95.6%

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

      6. log-rec 96.0%

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

      7. clear-num 95.6%

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

      8. clear-num 95.6%

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

      9. associate-*r* 95.6%

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

      10. associate-/l* 95.6%

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

      11. metadata-eval 95.6%

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

      12. clear-num 95.6%

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

      13. associate-/l/ 95.6%

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

      14. clear-num 96.0%

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

      15. div-inv 96.0%

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

      16. clear-num 96.0%

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

      17. div-inv 96.0%

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

      18. metadata-eval 96.0%

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

   10. Applied egg-rr 96.0%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{f \cdot \frac{\pi}{4}} + e^{f \cdot \frac{-\pi}{4}}}{e^{f \cdot \frac{\pi}{4}} - e^{f \cdot \frac{-\pi}{4}}} \leq 400000:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(\pi \cdot f\right)} + e^{0.25 \cdot \left(\pi \cdot f\right)}}{e^{0.25 \cdot \left(\pi \cdot f\right)} - e^{-0.25 \cdot \left(\pi \cdot f\right)}}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\pi} \cdot \left(--4\right)\\ \end{array} \]

Alternative 4: 95.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.3%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
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. Simplified 7.3%

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

4. Taylor expanded in f around 0 92.7%

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

   1. expm1-log1p-u 0.2%

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

   2. expm1-udef 0.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left(\frac{2}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}\right)} - 1} \]

   3. distribute-rgt-out-- 0.2%

      \[\leadsto e^{\mathsf{log1p}\left(\log \left(\frac{2}{f \cdot \color{blue}{\left(\pi \cdot \left(0.25 - -0.25\right)\right)}}\right) \cdot \frac{-4}{\pi}\right)} - 1 \]

   4. metadata-eval 0.2%

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

6. Applied egg-rr 0.2%

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

   1. expm1-def 0.2%

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

   2. expm1-log1p 92.7%

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

   3. associate-*r/ 92.9%

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

   4. *-commutative 92.9%

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

   5. *-lft-identity 92.9%

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

   6. times-frac 92.9%

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

   7. metadata-eval 92.9%

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

8. Simplified 92.9%

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

   1. associate-/l/ 92.9%

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

   2. clear-num 92.9%

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

   3. metadata-eval 92.9%

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

   4. associate-/l* 92.9%

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

   5. associate-*r* 92.9%

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

   6. log-rec 93.3%

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

   7. clear-num 92.9%

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

   8. clear-num 92.9%

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

   9. associate-*r* 92.9%

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

   10. associate-/l* 92.9%

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

   11. metadata-eval 92.9%

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

   12. clear-num 92.9%

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

   13. associate-/l/ 92.9%

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

   14. clear-num 93.3%

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

   15. div-inv 93.3%

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

   16. clear-num 93.3%

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

   17. div-inv 93.3%

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

   18. metadata-eval 93.3%

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

10. Applied egg-rr 93.3%

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

11. Final simplification 93.3%

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

Alternative 5: 95.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(N[(4.0 / f), $MachinePrecision] / Pi), $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. Simplified 7.3%

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

4. Taylor expanded in f around 0 92.7%

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

   1. expm1-log1p-u 0.2%

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

   2. expm1-udef 0.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left(\frac{2}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}\right)} - 1} \]

   3. distribute-rgt-out-- 0.2%

      \[\leadsto e^{\mathsf{log1p}\left(\log \left(\frac{2}{f \cdot \color{blue}{\left(\pi \cdot \left(0.25 - -0.25\right)\right)}}\right) \cdot \frac{-4}{\pi}\right)} - 1 \]

   4. metadata-eval 0.2%

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

6. Applied egg-rr 0.2%

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

   1. expm1-def 0.2%

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

   2. expm1-log1p 92.7%

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

   3. associate-*r/ 92.9%

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

   4. *-commutative 92.9%

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

   5. *-lft-identity 92.9%

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

   6. times-frac 92.9%

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

   7. metadata-eval 92.9%

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

8. Simplified 92.9%

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

9. Taylor expanded in f around 0 92.9%

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

    1. associate-/r* 92.9%

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

11. Simplified 92.9%

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

12. Final simplification 92.9%

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

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)))))))))