VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.0% → 96.8%

Time: 27.4s

Alternatives: 10

Speedup: 4.9×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{\frac{\pi}{4}}\right)}^{f}\\ t_1 := e^{f \cdot \frac{\pi}{-4}}\\ t_2 := e^{\frac{\pi}{4} \cdot f}\\ t_3 := {\left(e^{\frac{\pi}{-4}}\right)}^{f}\\ \mathbf{if}\;\frac{t\_2 + t\_1}{t\_2 - t\_1} \leq 10000:\\ \;\;\;\;\frac{\log \left(\frac{t\_3 + t\_0}{t\_0 - t\_3}\right)}{\frac{\pi}{-4}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (pow (exp (/ PI 4.0)) f))
        (t_1 (exp (* f (/ PI (- 4.0)))))
        (t_2 (exp (* (/ PI 4.0) f)))
        (t_3 (pow (exp (/ PI -4.0)) f)))
   (if (<= (/ (+ t_2 t_1) (- t_2 t_1)) 10000.0)
     (/ (log (/ (+ t_3 t_0) (- t_0 t_3))) (/ PI -4.0))
     (* -4.0 (/ (- (log (/ 4.0 PI)) (log f)) PI)))))

C

double code(double f) {
	double t_0 = pow(exp((((double) M_PI) / 4.0)), f);
	double t_1 = exp((f * (((double) M_PI) / -4.0)));
	double t_2 = exp(((((double) M_PI) / 4.0) * f));
	double t_3 = pow(exp((((double) M_PI) / -4.0)), f);
	double tmp;
	if (((t_2 + t_1) / (t_2 - t_1)) <= 10000.0) {
		tmp = log(((t_3 + t_0) / (t_0 - t_3))) / (((double) M_PI) / -4.0);
	} else {
		tmp = -4.0 * ((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = Math.pow(Math.exp((Math.PI / 4.0)), f);
	double t_1 = Math.exp((f * (Math.PI / -4.0)));
	double t_2 = Math.exp(((Math.PI / 4.0) * f));
	double t_3 = Math.pow(Math.exp((Math.PI / -4.0)), f);
	double tmp;
	if (((t_2 + t_1) / (t_2 - t_1)) <= 10000.0) {
		tmp = Math.log(((t_3 + t_0) / (t_0 - t_3))) / (Math.PI / -4.0);
	} else {
		tmp = -4.0 * ((Math.log((4.0 / Math.PI)) - Math.log(f)) / Math.PI);
	}
	return tmp;
}

Python

def code(f):
	t_0 = math.pow(math.exp((math.pi / 4.0)), f)
	t_1 = math.exp((f * (math.pi / -4.0)))
	t_2 = math.exp(((math.pi / 4.0) * f))
	t_3 = math.pow(math.exp((math.pi / -4.0)), f)
	tmp = 0
	if ((t_2 + t_1) / (t_2 - t_1)) <= 10000.0:
		tmp = math.log(((t_3 + t_0) / (t_0 - t_3))) / (math.pi / -4.0)
	else:
		tmp = -4.0 * ((math.log((4.0 / math.pi)) - math.log(f)) / math.pi)
	return tmp

Julia

function code(f)
	t_0 = exp(Float64(pi / 4.0)) ^ f
	t_1 = exp(Float64(f * Float64(pi / Float64(-4.0))))
	t_2 = exp(Float64(Float64(pi / 4.0) * f))
	t_3 = exp(Float64(pi / -4.0)) ^ f
	tmp = 0.0
	if (Float64(Float64(t_2 + t_1) / Float64(t_2 - t_1)) <= 10000.0)
		tmp = Float64(log(Float64(Float64(t_3 + t_0) / Float64(t_0 - t_3))) / Float64(pi / -4.0));
	else
		tmp = Float64(-4.0 * Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	t_0 = exp((pi / 4.0)) ^ f;
	t_1 = exp((f * (pi / -4.0)));
	t_2 = exp(((pi / 4.0) * f));
	t_3 = exp((pi / -4.0)) ^ f;
	tmp = 0.0;
	if (((t_2 + t_1) / (t_2 - t_1)) <= 10000.0)
		tmp = log(((t_3 + t_0) / (t_0 - t_3))) / (pi / -4.0);
	else
		tmp = -4.0 * ((log((4.0 / pi)) - log(f)) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := Block[{t$95$0 = N[Power[N[Exp[N[(Pi / 4.0), $MachinePrecision]], $MachinePrecision], f], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(f * N[(Pi / (-4.0)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Exp[N[(Pi / -4.0), $MachinePrecision]], $MachinePrecision], f], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 + t$95$1), $MachinePrecision] / N[(t$95$2 - t$95$1), $MachinePrecision]), $MachinePrecision], 10000.0], N[(N[Log[N[(N[(t$95$3 + t$95$0), $MachinePrecision] / N[(t$95$0 - t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi / -4.0), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $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))))) < 1e4

   1. Initial program 76.7%

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

      1. distribute-lft-neg-in 76.7%

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

      2. distribute-neg-frac2 76.7%

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

      3. associate-*l/ 76.9%

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

      4. *-lft-identity 76.9%

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

   3. Simplified 77.0%

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

   if 1e4 < (/.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 4.3%

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

      1. distribute-lft-neg-in 4.3%

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

      2. distribute-neg-frac2 4.3%

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

      3. associate-*l/ 4.3%

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

      4. *-lft-identity 4.3%

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

   3. Simplified 4.3%

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

   5. Taylor expanded in f around 0 97.6%

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

      1. *-commutative 97.6%

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

      2. associate-*l/ 97.6%

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

      3. associate-/l* 97.5%

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

      4. mul-1-neg 97.5%

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

      5. unsub-neg 97.5%

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

      6. distribute-rgt-out-- 97.5%

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

      7. metadata-eval 97.5%

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

      8. *-commutative 97.5%

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

      9. associate-/r* 97.5%

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

      10. metadata-eval 97.5%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in f around 0 97.6%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{e^{\frac{\pi}{4} \cdot f} - e^{f \cdot \frac{\pi}{-4}}} \leq 10000:\\ \;\;\;\;\frac{\log \left(\frac{{\left(e^{\frac{\pi}{-4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{f}}{{\left(e^{\frac{\pi}{4}}\right)}^{f} - {\left(e^{\frac{\pi}{-4}}\right)}^{f}}\right)}{\frac{\pi}{-4}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, {\left(\pi \cdot f\right)}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right) - \log \left(2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ 1.0 (/ PI 4.0))
  (-
   (log
    (fma
     f
     (* PI 0.5)
     (fma
      (pow f 3.0)
      (* (pow PI 3.0) 0.005208333333333333)
      (fma
       (pow f 5.0)
       (* (pow PI 5.0) 1.6276041666666666e-5)
       (* (pow (* PI f) 7.0) 2.422030009920635e-8)))))
   (log (* 2.0 (cosh (* (* PI 0.25) f)))))))

C

double code(double f) {
	return (1.0 / (((double) M_PI) / 4.0)) * (log(fma(f, (((double) M_PI) * 0.5), fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), (pow((((double) M_PI) * f), 7.0) * 2.422030009920635e-8))))) - log((2.0 * cosh(((((double) M_PI) * 0.25) * f)))));
}

Julia

function code(f)
	return Float64(Float64(1.0 / Float64(pi / 4.0)) * Float64(log(fma(f, Float64(pi * 0.5), fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), Float64((Float64(pi * f) ^ 7.0) * 2.422030009920635e-8))))) - log(Float64(2.0 * cosh(Float64(Float64(pi * 0.25) * f))))))
end

Wolfram

code[f_] := N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[Log[N[(f * N[(Pi * 0.5), $MachinePrecision] + N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision] + N[(N[Power[N[(Pi * f), $MachinePrecision], 7.0], $MachinePrecision] * 2.422030009920635e-8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[N[(2.0 * N[Cosh[N[(N[(Pi * 0.25), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in f around 0 95.0%

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

   1. fma-define 95.0%

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

   2. distribute-rgt-out-- 95.0%

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

   3. metadata-eval 95.0%

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

   4. fma-define 95.0%

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

   5. distribute-rgt-out-- 95.0%

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

   6. metadata-eval 95.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

5. Simplified 95.0%

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

   1. log-div 95.3%

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

   2. cosh-undef 95.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \color{blue}{\left(2 \cdot \cosh \left(\frac{\pi}{4} \cdot f\right)\right)} - \log \left(\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right)\right)\right) \]

   3. div-inv 95.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(2 \cdot \cosh \left(\color{blue}{\left(\pi \cdot \frac{1}{4}\right)} \cdot f\right)\right) - \log \left(\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right)\right)\right) \]

   4. metadata-eval 95.3%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(2 \cdot \cosh \left(\left(\pi \cdot \color{blue}{0.25}\right) \cdot f\right)\right) - \log \left(\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right)\right)\right) \]

7. Applied egg-rr 95.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)\right) - \log \left(\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, {\left(\pi \cdot f\right)}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right)\right)} \]

8. Final simplification 95.3%

   \[\leadsto \frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, {\left(\pi \cdot f\right)}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right) - \log \left(2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)\right)\right) \]
9. Add Preprocessing

Alternative 3: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{0.25 \cdot \left(\pi \cdot f\right)}\\ t_1 := e^{f \cdot \frac{\pi}{-4}}\\ t_2 := e^{\frac{\pi}{4} \cdot f}\\ t_3 := e^{-0.25 \cdot \left(\pi \cdot f\right)}\\ \mathbf{if}\;\frac{t\_2 + t\_1}{t\_2 - t\_1} \leq 10000:\\ \;\;\;\;\frac{\log \left(\frac{t\_3 + t\_0}{t\_0 - t\_3}\right)}{\frac{\pi}{-4}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (* 0.25 (* PI f))))
        (t_1 (exp (* f (/ PI (- 4.0)))))
        (t_2 (exp (* (/ PI 4.0) f)))
        (t_3 (exp (* -0.25 (* PI f)))))
   (if (<= (/ (+ t_2 t_1) (- t_2 t_1)) 10000.0)
     (/ (log (/ (+ t_3 t_0) (- t_0 t_3))) (/ PI -4.0))
     (* -4.0 (/ (- (log (/ 4.0 PI)) (log f)) PI)))))

C

double code(double f) {
	double t_0 = exp((0.25 * (((double) M_PI) * f)));
	double t_1 = exp((f * (((double) M_PI) / -4.0)));
	double t_2 = exp(((((double) M_PI) / 4.0) * f));
	double t_3 = exp((-0.25 * (((double) M_PI) * f)));
	double tmp;
	if (((t_2 + t_1) / (t_2 - t_1)) <= 10000.0) {
		tmp = log(((t_3 + t_0) / (t_0 - t_3))) / (((double) M_PI) / -4.0);
	} else {
		tmp = -4.0 * ((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = Math.exp((0.25 * (Math.PI * f)));
	double t_1 = Math.exp((f * (Math.PI / -4.0)));
	double t_2 = Math.exp(((Math.PI / 4.0) * f));
	double t_3 = Math.exp((-0.25 * (Math.PI * f)));
	double tmp;
	if (((t_2 + t_1) / (t_2 - t_1)) <= 10000.0) {
		tmp = Math.log(((t_3 + t_0) / (t_0 - t_3))) / (Math.PI / -4.0);
	} else {
		tmp = -4.0 * ((Math.log((4.0 / Math.PI)) - Math.log(f)) / Math.PI);
	}
	return tmp;
}

Python

def code(f):
	t_0 = math.exp((0.25 * (math.pi * f)))
	t_1 = math.exp((f * (math.pi / -4.0)))
	t_2 = math.exp(((math.pi / 4.0) * f))
	t_3 = math.exp((-0.25 * (math.pi * f)))
	tmp = 0
	if ((t_2 + t_1) / (t_2 - t_1)) <= 10000.0:
		tmp = math.log(((t_3 + t_0) / (t_0 - t_3))) / (math.pi / -4.0)
	else:
		tmp = -4.0 * ((math.log((4.0 / math.pi)) - math.log(f)) / math.pi)
	return tmp

Julia

function code(f)
	t_0 = exp(Float64(0.25 * Float64(pi * f)))
	t_1 = exp(Float64(f * Float64(pi / Float64(-4.0))))
	t_2 = exp(Float64(Float64(pi / 4.0) * f))
	t_3 = exp(Float64(-0.25 * Float64(pi * f)))
	tmp = 0.0
	if (Float64(Float64(t_2 + t_1) / Float64(t_2 - t_1)) <= 10000.0)
		tmp = Float64(log(Float64(Float64(t_3 + t_0) / Float64(t_0 - t_3))) / Float64(pi / -4.0));
	else
		tmp = Float64(-4.0 * Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	t_0 = exp((0.25 * (pi * f)));
	t_1 = exp((f * (pi / -4.0)));
	t_2 = exp(((pi / 4.0) * f));
	t_3 = exp((-0.25 * (pi * f)));
	tmp = 0.0;
	if (((t_2 + t_1) / (t_2 - t_1)) <= 10000.0)
		tmp = log(((t_3 + t_0) / (t_0 - t_3))) / (pi / -4.0);
	else
		tmp = -4.0 * ((log((4.0 / pi)) - log(f)) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := Block[{t$95$0 = N[Exp[N[(0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(f * N[(Pi / (-4.0)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(-0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 + t$95$1), $MachinePrecision] / N[(t$95$2 - t$95$1), $MachinePrecision]), $MachinePrecision], 10000.0], N[(N[Log[N[(N[(t$95$3 + t$95$0), $MachinePrecision] / N[(t$95$0 - t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi / -4.0), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $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))))) < 1e4

   1. Initial program 76.7%

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

      1. distribute-lft-neg-in 76.7%

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

      2. distribute-neg-frac2 76.7%

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

      3. associate-*l/ 76.9%

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

      4. *-lft-identity 76.9%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in f around inf 76.9%

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

   if 1e4 < (/.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 4.3%

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

      1. distribute-lft-neg-in 4.3%

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

      2. distribute-neg-frac2 4.3%

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

      3. associate-*l/ 4.3%

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

      4. *-lft-identity 4.3%

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

   3. Simplified 4.3%

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

   5. Taylor expanded in f around 0 97.6%

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

      1. *-commutative 97.6%

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

      2. associate-*l/ 97.6%

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

      3. associate-/l* 97.5%

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

      4. mul-1-neg 97.5%

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

      5. unsub-neg 97.5%

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

      6. distribute-rgt-out-- 97.5%

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

      7. metadata-eval 97.5%

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

      8. *-commutative 97.5%

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

      9. associate-/r* 97.5%

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

      10. metadata-eval 97.5%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in f around 0 97.6%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{e^{\frac{\pi}{4} \cdot f} - e^{f \cdot \frac{\pi}{-4}}} \leq 10000:\\ \;\;\;\;\frac{\log \left(\frac{e^{-0.25 \cdot \left(\pi \cdot f\right)} + e^{0.25 \cdot \left(\pi \cdot f\right)}}{e^{0.25 \cdot \left(\pi \cdot f\right)} - e^{-0.25 \cdot \left(\pi \cdot f\right)}}\right)}{\frac{\pi}{-4}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{\pi \cdot f}\right)\right)}{-\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (* 4.0 (log (fma f (* PI 0.08333333333333333) (/ 4.0 (* PI f))))) (- PI)))

C

double code(double f) {
	return (4.0 * log(fma(f, (((double) M_PI) * 0.08333333333333333), (4.0 / (((double) M_PI) * f))))) / -((double) M_PI);
}

Julia

function code(f)
	return Float64(Float64(4.0 * log(fma(f, Float64(pi * 0.08333333333333333), Float64(4.0 / Float64(pi * f))))) / Float64(-pi))
end

Wolfram

code[f_] := N[(N[(4.0 * N[Log[N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-Pi)), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in f around 0 94.6%

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

5. Simplified 94.6%

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

   1. add-sqr-sqrt 94.2%

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

   2. pow2 94.2%

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

7. Applied egg-rr 94.2%

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

   1. unpow2 94.2%

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

   2. add-sqr-sqrt 94.7%

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

   3. *-commutative 94.7%

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

   4. fma-define 94.7%

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

   5. associate-*l* 94.7%

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

   6. metadata-eval 94.7%

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

   7. associate-/l/ 94.7%

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

9. Applied egg-rr 94.7%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \pi \cdot -0.041666666666666664\right), \frac{4}{\pi \cdot f}\right)\right)}{\pi}} \]
10. Step-by-step derivation

    1. associate-*r/ 94.7%

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

    2. fma-undefine 94.7%

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

    3. distribute-lft-out 94.7%

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

    4. metadata-eval 94.7%

       \[\leadsto -\frac{4 \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.08333333333333333}, \frac{4}{\pi \cdot f}\right)\right)}{\pi} \]

11. Simplified 94.7%

    \[\leadsto -\color{blue}{\frac{4 \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{\pi \cdot f}\right)\right)}{\pi}} \]

12. Final simplification 94.7%

    \[\leadsto \frac{4 \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{\pi \cdot f}\right)\right)}{-\pi} \]
13. Add Preprocessing

Alternative 5: 95.7% 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 7.1%

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

   1. distribute-lft-neg-in 7.1%

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

   2. distribute-neg-frac2 7.1%

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

   3. associate-*l/ 7.1%

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

   4. *-lft-identity 7.1%

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

3. Simplified 7.1%

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

5. Taylor expanded in f around 0 94.6%

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

   1. *-commutative 94.6%

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

   2. associate-*l/ 94.6%

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

   3. associate-/l* 94.5%

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

   4. mul-1-neg 94.5%

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

   5. unsub-neg 94.5%

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

   6. distribute-rgt-out-- 94.5%

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

   7. metadata-eval 94.5%

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

   8. *-commutative 94.5%

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

   9. associate-/r* 94.5%

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

   10. metadata-eval 94.5%

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

7. Simplified 94.5%

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

8. Taylor expanded in f around 0 94.6%

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

9. Final simplification 94.6%

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

Alternative 6: 95.5% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in f around 0 94.1%

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

   1. associate-/l/ 94.1%

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

   2. distribute-rgt-out-- 94.1%

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

   3. metadata-eval 94.1%

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

   4. *-commutative 94.1%

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

   5. associate-/r* 94.1%

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

   6. metadata-eval 94.1%

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

5. Simplified 94.1%

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

   1. clear-num 94.1%

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

   2. log-rec 94.5%

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

7. Applied egg-rr 94.5%

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

8. Final simplification 94.5%

   \[\leadsto \frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{f}{\frac{4}{\pi}}\right) \]
9. Add Preprocessing

Alternative 7: 1.7% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. distribute-lft-neg-in 7.1%

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

   2. distribute-neg-frac2 7.1%

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

   3. associate-*l/ 7.1%

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

   4. *-lft-identity 7.1%

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

3. Simplified 7.1%

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

5. Taylor expanded in f around 0 94.6%

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

   1. *-commutative 94.6%

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

   2. associate-*l/ 94.6%

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

   3. associate-/l* 94.5%

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

   4. mul-1-neg 94.5%

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

   5. unsub-neg 94.5%

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

   6. distribute-rgt-out-- 94.5%

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

   7. metadata-eval 94.5%

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

   8. *-commutative 94.5%

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

   9. associate-/r* 94.5%

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

   10. metadata-eval 94.5%

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

7. Simplified 94.5%

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

   1. add-cube-cbrt 93.0%

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

   2. pow3 93.0%

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

   3. diff-log 92.6%

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

   4. associate-/l/ 92.6%

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

   5. *-commutative 92.6%

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

9. Applied egg-rr 92.6%

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

    1. rem-cube-cbrt 94.1%

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

    2. diff-log 94.4%

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

    3. *-commutative 94.4%

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

    4. add-sqr-sqrt 0.0%

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

    5. sqrt-unprod 1.7%

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

    6. frac-times 1.7%

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

    7. metadata-eval 1.7%

       \[\leadsto \sqrt{\frac{\color{blue}{16}}{\pi \cdot \pi}} \cdot \left(\log 4 - \log \left(\pi \cdot f\right)\right) \]

    8. metadata-eval 1.7%

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

    9. frac-times 1.7%

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

    10. sqrt-unprod 1.7%

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

    11. add-sqr-sqrt 1.7%

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

    12. clear-num 1.7%

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

    13. associate-*l/ 1.7%

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

    14. div-inv 1.7%

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

    15. metadata-eval 1.7%

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

11. Applied egg-rr 1.7%

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

12. Taylor expanded in f around 0 1.7%

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

    1. *-commutative 1.7%

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

14. Simplified 1.7%

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

15. Final simplification 1.7%

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

Alternative 8: 95.5% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. distribute-lft-neg-in 7.1%

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

   2. distribute-neg-frac2 7.1%

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

   3. associate-*l/ 7.1%

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

   4. *-lft-identity 7.1%

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

3. Simplified 7.1%

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

5. Taylor expanded in f around 0 94.6%

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

   1. *-commutative 94.6%

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

   2. associate-*l/ 94.6%

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

   3. associate-/l* 94.5%

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

   4. mul-1-neg 94.5%

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

   5. unsub-neg 94.5%

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

   6. distribute-rgt-out-- 94.5%

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

   7. metadata-eval 94.5%

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

   8. *-commutative 94.5%

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

   9. associate-/r* 94.5%

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

   10. metadata-eval 94.5%

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

7. Simplified 94.5%

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

   1. add-cube-cbrt 93.0%

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

   2. pow3 93.0%

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

   3. diff-log 92.6%

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

   4. associate-/l/ 92.6%

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

   5. *-commutative 92.6%

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

9. Applied egg-rr 92.6%

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

10. Taylor expanded in f around 0 94.6%

    \[\leadsto \color{blue}{-4 \cdot \left({1}^{0.3333333333333333} \cdot \frac{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}{\pi}\right)} \]
11. Step-by-step derivation

    1. pow-base-1 94.6%

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

    2. associate-*r/ 94.6%

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

    3. *-lft-identity 94.6%

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

    4. associate-*r/ 94.6%

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

    5. mul-1-neg 94.6%

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

    6. sub-neg 94.6%

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

    7. log-div 94.6%

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

    8. associate--r+ 94.5%

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

    9. log-prod 94.5%

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

    10. log-div 94.2%

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

    11. associate-/l/ 94.2%

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

12. Simplified 94.1%

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

13. Final simplification 94.1%

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

Alternative 9: 95.6% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. distribute-lft-neg-in 7.1%

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

   2. distribute-neg-frac2 7.1%

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

   3. associate-*l/ 7.1%

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

   4. *-lft-identity 7.1%

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

3. Simplified 7.1%

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

5. Taylor expanded in f around 0 94.6%

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

   1. *-commutative 94.6%

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

   2. associate-*l/ 94.6%

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

   3. associate-/l* 94.5%

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

   4. mul-1-neg 94.5%

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

   5. unsub-neg 94.5%

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

   6. distribute-rgt-out-- 94.5%

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

   7. metadata-eval 94.5%

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

   8. *-commutative 94.5%

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

   9. associate-/r* 94.5%

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

   10. metadata-eval 94.5%

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

7. Simplified 94.5%

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

   1. associate-*r/ 94.6%

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

   2. diff-log 94.2%

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

   3. associate-/l/ 94.2%

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

   4. *-commutative 94.2%

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

9. Applied egg-rr 94.2%

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

10. Final simplification 94.2%

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

Alternative 10: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log 0.125}{\pi} \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ (log 0.125) PI) (- 4.0)))

C

double code(double f) {
	return (log(0.125) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return (Math.log(0.125) / Math.PI) * -4.0;
}

Python

def code(f):
	return (math.log(0.125) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(log(0.125) / pi) * Float64(-4.0))
end

MATLAB

function tmp = code(f)
	tmp = (log(0.125) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[Log[0.125], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

4. Applied egg-rr 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{16}}\right) \]

5. Taylor expanded in f around 0 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\color{blue}{\left(1 + 0.25 \cdot \left(f \cdot \pi\right)\right)} + e^{-\frac{\pi}{4} \cdot f}}{16}\right) \]
6. Step-by-step derivation

   1. +-commutative 1.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\color{blue}{\left(0.25 \cdot \left(f \cdot \pi\right) + 1\right)} + e^{-\frac{\pi}{4} \cdot f}}{16}\right) \]

   2. associate-*r* 1.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\left(\color{blue}{\left(0.25 \cdot f\right) \cdot \pi} + 1\right) + e^{-\frac{\pi}{4} \cdot f}}{16}\right) \]

7. Simplified 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\color{blue}{\left(\left(0.25 \cdot f\right) \cdot \pi + 1\right)} + e^{-\frac{\pi}{4} \cdot f}}{16}\right) \]

8. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.125}{\pi}} \]

9. Final simplification 1.6%

   \[\leadsto \frac{\log 0.125}{\pi} \cdot \left(-4\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024043 
(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)))))))))