VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 96.6%

Time: 18.0s

Alternatives: 9

Speedup: 9.9×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{-\log \left(\frac{e^{0.25 \cdot \left(f \cdot \left(-\pi\right)\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)\right)}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  4.0
  (/
   (-
    (log
     (/
      (+ (exp (* 0.25 (* f (- PI)))) (exp (* 0.25 (* f PI))))
      (fma
       f
       (* PI 0.5)
       (fma
        (pow f 3.0)
        (* (pow PI 3.0) 0.005208333333333333)
        (* (pow f 5.0) (* (pow PI 5.0) 1.6276041666666666e-5)))))))
   PI)))

C

double code(double f) {
	return 4.0 * (-log(((exp((0.25 * (f * -((double) M_PI)))) + exp((0.25 * (f * ((double) M_PI))))) / fma(f, (((double) M_PI) * 0.5), fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), (pow(f, 5.0) * (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5)))))) / ((double) M_PI));
}

Julia

function code(f)
	return Float64(4.0 * Float64(Float64(-log(Float64(Float64(exp(Float64(0.25 * Float64(f * Float64(-pi)))) + exp(Float64(0.25 * Float64(f * pi)))) / fma(f, Float64(pi * 0.5), fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), Float64((f ^ 5.0) * Float64((pi ^ 5.0) * 1.6276041666666666e-5))))))) / pi))
end

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in f around inf 6.8%

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

3. Taylor expanded in f around 0 97.7%

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

   1. fma-def 97.7%

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

   2. distribute-rgt-out-- 97.7%

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

   3. metadata-eval 97.7%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right)}{\pi} \]

   4. fma-def 97.7%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\mathsf{fma}\left({f}^{3}, 0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right)}\right)}{\pi} \]

   5. distribute-rgt-out-- 97.7%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, \color{blue}{{\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)\right)}\right)}{\pi} \]

   6. metadata-eval 97.7%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)\right)}\right)}{\pi} \]

   7. distribute-rgt-out-- 97.7%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \color{blue}{\left({\pi}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} - -8.138020833333333 \cdot 10^{-6}\right)\right)}\right)\right)}\right)}{\pi} \]

   8. metadata-eval 97.7%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \left({\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}\right)\right)\right)}\right)}{\pi} \]

5. Simplified 97.7%

   \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\color{blue}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)\right)}}\right)}{\pi} \]

6. Final simplification 97.7%

   \[\leadsto 4 \cdot \frac{-\log \left(\frac{e^{0.25 \cdot \left(f \cdot \left(-\pi\right)\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)\right)}\right)}{\pi} \]

Alternative 2: 96.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in f around 0 97.4%

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

4. Simplified 97.4%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, 2 \cdot \pi, \frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}} \cdot -2\right), \frac{\frac{4}{\pi}}{f}\right)\right)} \]
5. Step-by-step derivation

   1. clear-num 97.4%

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

   2. expm1-log1p-u 96.3%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{4}{\pi} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, 2 \cdot \pi, \frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}} \cdot -2\right), \frac{\frac{4}{\pi}}{f}\right)\right)\right)\right)} \]

   3. expm1-udef 96.3%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{4}{\pi} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, 2 \cdot \pi, \frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}} \cdot -2\right), \frac{\frac{4}{\pi}}{f}\right)\right)\right)} - 1\right)} \]

6. Applied egg-rr 96.3%

   \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{4}{\pi} \cdot \log \left(\mathsf{fma}\left(f, -0.041666666666666664 \cdot \pi + 0.125 \cdot \pi, \frac{4}{\pi \cdot f}\right)\right)\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 96.3%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{4}{\pi} \cdot \log \left(\mathsf{fma}\left(f, -0.041666666666666664 \cdot \pi + 0.125 \cdot \pi, \frac{4}{\pi \cdot f}\right)\right)\right)\right)} \]

   2. expm1-log1p 97.4%

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

   3. associate-*l/ 97.5%

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

   4. *-commutative 97.5%

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

   5. associate-*l/ 97.5%

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

   6. *-commutative 97.5%

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

   7. distribute-rgt-out 97.5%

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

   8. metadata-eval 97.5%

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

8. Simplified 97.5%

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

9. Final simplification 97.5%

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

Alternative 3: 95.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in f around 0 97.0%

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

   1. associate-*r/ 97.0%

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

   2. associate-/l* 96.9%

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

   3. associate-/r/ 97.0%

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

   4. mul-1-neg 97.0%

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

   5. unsub-neg 97.0%

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

   6. distribute-rgt-out-- 97.0%

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

   7. *-commutative 97.0%

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

   8. associate-/r* 97.0%

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

   9. metadata-eval 97.0%

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

   10. metadata-eval 97.0%

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

4. Simplified 97.0%

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

5. Final simplification 97.0%

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

Alternative 4: 95.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in f around inf 6.8%

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

3. Taylor expanded in f around 0 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-/r* 97.0%

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

   3. distribute-rgt-out-- 97.0%

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

   4. metadata-eval 97.0%

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

5. Simplified 97.0%

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

6. Taylor expanded in f around 0 97.0%

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

   1. associate-/l/ 97.0%

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

8. Simplified 97.0%

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

9. Final simplification 97.0%

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

Alternative 5: 17.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log 0.005224609375 \cdot -16}{\pi} \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (/ (* (log 0.005224609375) -16.0) PI) (- 4.0)))

C

double code(double f) {
	return ((log(0.005224609375) * -16.0) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return ((Math.log(0.005224609375) * -16.0) / Math.PI) * -4.0;
}

Python

def code(f):
	return ((math.log(0.005224609375) * -16.0) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(Float64(log(0.005224609375) * -16.0) / pi) * Float64(-4.0))
end

MATLAB

function tmp = code(f)
	tmp = ((log(0.005224609375) * -16.0) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[(N[Log[0.005224609375], $MachinePrecision] * -16.0), $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in f around inf 6.8%

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

3. Taylor expanded in f around 0 97.7%

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

   1. fma-def 97.7%

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

   2. distribute-rgt-out-- 97.7%

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

   3. metadata-eval 97.7%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right)}{\pi} \]

   4. fma-def 97.7%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\mathsf{fma}\left({f}^{3}, 0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right)}\right)}{\pi} \]

   5. distribute-rgt-out-- 97.7%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, \color{blue}{{\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)\right)}\right)}{\pi} \]

   6. metadata-eval 97.7%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)\right)}\right)}{\pi} \]

   7. distribute-rgt-out-- 97.7%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \color{blue}{\left({\pi}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} - -8.138020833333333 \cdot 10^{-6}\right)\right)}\right)\right)}\right)}{\pi} \]

   8. metadata-eval 97.7%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \left({\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}\right)\right)\right)}\right)}{\pi} \]

5. Simplified 97.7%

   \[\leadsto -4 \cdot \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\color{blue}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)\right)}}\right)}{\pi} \]

7. Applied egg-rr 17.3%

   \[\leadsto -4 \cdot \frac{\color{blue}{\log 0.005224609375 \cdot -16}}{\pi} \]

8. Final simplification 17.3%

   \[\leadsto \frac{\log 0.005224609375 \cdot -16}{\pi} \cdot \left(-4\right) \]

Alternative 6: 16.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \log 0.001953125 \cdot \left(-\log 0.001953125\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (log 0.001953125) (- (log 0.001953125))))

C

double code(double f) {
	return log(0.001953125) * -log(0.001953125);
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = log(0.001953125d0) * -log(0.001953125d0)
end function

Java

public static double code(double f) {
	return Math.log(0.001953125) * -Math.log(0.001953125);
}

Python

def code(f):
	return math.log(0.001953125) * -math.log(0.001953125)

Julia

function code(f)
	return Float64(log(0.001953125) * Float64(-log(0.001953125)))
end

MATLAB

function tmp = code(f)
	tmp = log(0.001953125) * -log(0.001953125);
end

Wolfram

code[f_] := N[(N[Log[0.001953125], $MachinePrecision] * (-N[Log[0.001953125], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in f around 0 96.9%

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

   1. distribute-rgt-out-- 96.9%

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

   2. metadata-eval 96.9%

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

4. Simplified 96.9%

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

5. Taylor expanded in f around 0 96.9%

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

7. Applied egg-rr 0.0%

   \[\leadsto -\color{blue}{e^{\log \log 0.001953125 + \log \log 0.001953125}} \]
8. Step-by-step derivation

   1. exp-sum 0.0%

      \[\leadsto -\color{blue}{e^{\log \log 0.001953125} \cdot e^{\log \log 0.001953125}} \]

   2. rem-exp-log 0.0%

      \[\leadsto -\color{blue}{\log 0.001953125} \cdot e^{\log \log 0.001953125} \]

   3. rem-exp-log 16.1%

      \[\leadsto -\log 0.001953125 \cdot \color{blue}{\log 0.001953125} \]

9. Simplified 16.1%

   \[\leadsto -\color{blue}{\log 0.001953125 \cdot \log 0.001953125} \]

10. Final simplification 16.1%

    \[\leadsto \log 0.001953125 \cdot \left(-\log 0.001953125\right) \]

Alternative 7: 14.6% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ -\left|\log 0.001953125\right| \end{array} \]

FPCore

(FPCore (f) :precision binary64 (- (fabs (log 0.001953125))))

C

double code(double f) {
	return -fabs(log(0.001953125));
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = -abs(log(0.001953125d0))
end function

Java

public static double code(double f) {
	return -Math.abs(Math.log(0.001953125));
}

Python

def code(f):
	return -math.fabs(math.log(0.001953125))

Julia

function code(f)
	return Float64(-abs(log(0.001953125)))
end

MATLAB

function tmp = code(f)
	tmp = -abs(log(0.001953125));
end

Wolfram

code[f_] := (-N[Abs[N[Log[0.001953125], $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in f around 0 96.9%

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

   1. distribute-rgt-out-- 96.9%

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

   2. metadata-eval 96.9%

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

4. Simplified 96.9%

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

5. Taylor expanded in f around 0 96.9%

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

7. Applied egg-rr 14.7%

   \[\leadsto -\color{blue}{\left|\log 0.001953125\right|} \]

8. Final simplification 14.7%

   \[\leadsto -\left|\log 0.001953125\right| \]

Alternative 8: 13.4% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \frac{-\log 0.001953125}{-16} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (- (log 0.001953125)) -16.0))

C

double code(double f) {
	return -log(0.001953125) / -16.0;
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = -log(0.001953125d0) / (-16.0d0)
end function

Java

public static double code(double f) {
	return -Math.log(0.001953125) / -16.0;
}

Python

def code(f):
	return -math.log(0.001953125) / -16.0

Julia

function code(f)
	return Float64(Float64(-log(0.001953125)) / -16.0)
end

MATLAB

function tmp = code(f)
	tmp = -log(0.001953125) / -16.0;
end

Wolfram

code[f_] := N[((-N[Log[0.001953125], $MachinePrecision]) / -16.0), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in f around 0 96.9%

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

   1. distribute-rgt-out-- 96.9%

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

   2. metadata-eval 96.9%

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

4. Simplified 96.9%

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

5. Taylor expanded in f around 0 96.9%

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

7. Applied egg-rr 13.5%

   \[\leadsto -\color{blue}{\frac{\log 0.001953125}{-16}} \]

8. Final simplification 13.5%

   \[\leadsto \frac{-\log 0.001953125}{-16} \]

Alternative 9: 1.6% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ -\log 0.001953125 \end{array} \]

FPCore

(FPCore (f) :precision binary64 (- (log 0.001953125)))

C

double code(double f) {
	return -log(0.001953125);
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = -log(0.001953125d0)
end function

Java

public static double code(double f) {
	return -Math.log(0.001953125);
}

Python

def code(f):
	return -math.log(0.001953125)

Julia

function code(f)
	return Float64(-log(0.001953125))
end

MATLAB

function tmp = code(f)
	tmp = -log(0.001953125);
end

Wolfram

code[f_] := (-N[Log[0.001953125], $MachinePrecision])

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in f around 0 96.9%

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

   1. distribute-rgt-out-- 96.9%

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

   2. metadata-eval 96.9%

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

4. Simplified 96.9%

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

5. Taylor expanded in f around 0 96.9%

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

7. Applied egg-rr 1.6%

   \[\leadsto -\color{blue}{\frac{-16}{\frac{-16}{\log 0.001953125}}} \]
8. Step-by-step derivation

   1. associate-/r/ 1.6%

      \[\leadsto -\color{blue}{\frac{-16}{-16} \cdot \log 0.001953125} \]

   2. metadata-eval 1.6%

      \[\leadsto -\color{blue}{1} \cdot \log 0.001953125 \]

   3. *-lft-identity 1.6%

      \[\leadsto -\color{blue}{\log 0.001953125} \]

9. Simplified 1.6%

   \[\leadsto -\color{blue}{\log 0.001953125} \]

10. Final simplification 1.6%

    \[\leadsto -\log 0.001953125 \]

Reproduce

herbie shell --seed 2023336 
(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)))))))))