VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 96.4%

Time: 32.6s

Alternatives: 9

Speedup: 5.0×

• could not determine a ground truth

  1. f = -3.642076323437423e+37

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(f)
	return Float64(4.0 * Float64(Float64(-log(Float64(Float64(exp(Float64(0.25 * Float64(f * pi))) + exp(Float64(Float64(f * pi) * -0.25))) / fma(f, Float64(cbrt((pi ^ 3.0)) * 0.5), fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), Float64((pi ^ 5.0) * Float64(1.6276041666666666e-5 * (f ^ 5.0)))))))) / 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[(N[(f * Pi), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(f * N[(N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.5), $MachinePrecision] + N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(N[Power[Pi, 5.0], $MachinePrecision] * N[(1.6276041666666666e-5 * N[Power[f, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.2%

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

2. Taylor expanded in f around inf 6.2%

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

4. Simplified 6.2%

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

5. Taylor expanded in f around 0 95.8%

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

   1. fma-def 95.8%

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

   2. distribute-rgt-out-- 95.8%

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

   3. metadata-eval 95.8%

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

   4. fma-def 95.8%

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

   5. distribute-rgt-out-- 95.8%

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

   6. metadata-eval 95.8%

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

   7. *-commutative 95.8%

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

   8. distribute-rgt-out-- 95.8%

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

   9. metadata-eval 95.8%

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

   10. associate-*l* 95.8%

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

7. Simplified 95.8%

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

   1. add-cbrt-cube 95.9%

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

   2. unpow3 95.9%

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

9. Applied egg-rr 95.9%

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

10. Final simplification 95.9%

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

Alternative 2: 96.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(f)
	return Float64(4.0 * Float64(-Float64(log(Float64(Float64(exp(Float64(0.25 * Float64(f * pi))) + exp(Float64(Float64(f * pi) * -0.25))) / fma(f, Float64(pi * 0.5), fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), Float64((pi ^ 5.0) * Float64(1.6276041666666666e-5 * (f ^ 5.0))))))) / 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[(N[(f * Pi), $MachinePrecision] * -0.25), $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[Pi, 5.0], $MachinePrecision] * N[(1.6276041666666666e-5 * N[Power[f, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 6.2%

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

2. Taylor expanded in f around inf 6.2%

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

4. Simplified 6.2%

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

5. Taylor expanded in f around 0 95.8%

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

   1. fma-def 95.8%

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

   2. distribute-rgt-out-- 95.8%

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

   3. metadata-eval 95.8%

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

   4. fma-def 95.8%

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

   5. distribute-rgt-out-- 95.8%

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

   6. metadata-eval 95.8%

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

   7. *-commutative 95.8%

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

   8. distribute-rgt-out-- 95.8%

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

   9. metadata-eval 95.8%

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

   10. associate-*l* 95.8%

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

7. Simplified 95.8%

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

8. Final simplification 95.8%

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

Alternative 3: 96.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \pi \cdot -0.041666666666666664\right), \frac{4}{\mathsf{expm1}\left(\mathsf{log1p}\left(f \cdot \pi\right)\right)}\right)\right)}{\pi} \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/
   (log
    (fma
     f
     (fma PI 0.125 (* PI -0.041666666666666664))
     (/ 4.0 (expm1 (log1p (* f PI))))))
   PI)
  (- 4.0)))

C

double code(double f) {
	return (log(fma(f, fma(((double) M_PI), 0.125, (((double) M_PI) * -0.041666666666666664)), (4.0 / expm1(log1p((f * ((double) M_PI))))))) / ((double) M_PI)) * -4.0;
}

Julia

function code(f)
	return Float64(Float64(log(fma(f, fma(pi, 0.125, Float64(pi * -0.041666666666666664)), Float64(4.0 / expm1(log1p(Float64(f * pi)))))) / pi) * Float64(-4.0))
end

Wolfram

code[f_] := N[(N[(N[Log[N[(f * N[(Pi * 0.125 + N[(Pi * -0.041666666666666664), $MachinePrecision]), $MachinePrecision] + N[(4.0 / N[(Exp[N[Log[1 + N[(f * Pi), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision]

Derivation

1. Initial program 6.2%

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

2. Taylor expanded in f around inf 6.2%

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

4. Simplified 6.2%

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

5. Taylor expanded in f around 0 95.7%

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

7. Simplified 95.7%

   \[\leadsto -4 \cdot \frac{\log \color{blue}{\left(\mathsf{fma}\left(f, 0.125 \cdot \frac{{\pi}^{2}}{\pi} - \frac{0.010416666666666666 \cdot {\pi}^{3}}{0.25 \cdot {\pi}^{2}}, \frac{4}{\pi \cdot f}\right) + 0\right)}}{\pi} \]
8. Step-by-step derivation

   1. sub-neg 95.7%

      \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \color{blue}{0.125 \cdot \frac{{\pi}^{2}}{\pi} + \left(-\frac{0.010416666666666666 \cdot {\pi}^{3}}{0.25 \cdot {\pi}^{2}}\right)}, \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

   2. times-frac 95.7%

      \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, 0.125 \cdot \frac{{\pi}^{2}}{\pi} + \left(-\color{blue}{\frac{0.010416666666666666}{0.25} \cdot \frac{{\pi}^{3}}{{\pi}^{2}}}\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

   3. metadata-eval 95.7%

      \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, 0.125 \cdot \frac{{\pi}^{2}}{\pi} + \left(-\color{blue}{0.041666666666666664} \cdot \frac{{\pi}^{3}}{{\pi}^{2}}\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

9. Applied egg-rr 95.7%

   \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \color{blue}{0.125 \cdot \frac{{\pi}^{2}}{\pi} + \left(-0.041666666666666664 \cdot \frac{{\pi}^{3}}{{\pi}^{2}}\right)}, \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]
10. Step-by-step derivation

    1. *-commutative 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\frac{{\pi}^{2}}{\pi} \cdot 0.125} + \left(-0.041666666666666664 \cdot \frac{{\pi}^{3}}{{\pi}^{2}}\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    2. fma-udef 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{fma}\left(\frac{{\pi}^{2}}{\pi}, 0.125, -0.041666666666666664 \cdot \frac{{\pi}^{3}}{{\pi}^{2}}\right)}, \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    3. *-commutative 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{{\pi}^{2}}{\pi}, 0.125, -\color{blue}{\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.041666666666666664}\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    4. unpow2 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{\color{blue}{\pi \cdot \pi}}{\pi}, 0.125, -\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.041666666666666664\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    5. associate-/l* 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\color{blue}{\frac{\pi}{\frac{\pi}{\pi}}}, 0.125, -\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.041666666666666664\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    6. *-inverses 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{\pi}{\color{blue}{1}}, 0.125, -\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.041666666666666664\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    7. /-rgt-identity 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\color{blue}{\pi}, 0.125, -\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.041666666666666664\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    8. distribute-rgt-neg-in 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \color{blue}{\frac{{\pi}^{3}}{{\pi}^{2}} \cdot \left(-0.041666666666666664\right)}\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    9. cube-mult 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \frac{\color{blue}{\pi \cdot \left(\pi \cdot \pi\right)}}{{\pi}^{2}} \cdot \left(-0.041666666666666664\right)\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    10. unpow2 95.7%

        \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \frac{\pi \cdot \color{blue}{{\pi}^{2}}}{{\pi}^{2}} \cdot \left(-0.041666666666666664\right)\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    11. associate-/l* 95.7%

        \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \color{blue}{\frac{\pi}{\frac{{\pi}^{2}}{{\pi}^{2}}}} \cdot \left(-0.041666666666666664\right)\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    12. *-inverses 95.7%

        \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \frac{\pi}{\color{blue}{1}} \cdot \left(-0.041666666666666664\right)\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    13. /-rgt-identity 95.7%

        \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \color{blue}{\pi} \cdot \left(-0.041666666666666664\right)\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    14. metadata-eval 95.7%

        \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \pi \cdot \color{blue}{-0.041666666666666664}\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

11. Simplified 95.7%

    \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{fma}\left(\pi, 0.125, \pi \cdot -0.041666666666666664\right)}, \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]
12. Step-by-step derivation

    1. *-commutative 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \pi \cdot -0.041666666666666664\right), \frac{4}{\color{blue}{f \cdot \pi}}\right) + 0\right)}{\pi} \]

    2. expm1-log1p-u 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \pi \cdot -0.041666666666666664\right), \frac{4}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(f \cdot \pi\right)\right)}}\right) + 0\right)}{\pi} \]

13. Applied egg-rr 95.7%

    \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \pi \cdot -0.041666666666666664\right), \frac{4}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(f \cdot \pi\right)\right)}}\right) + 0\right)}{\pi} \]

14. Final simplification 95.7%

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

Alternative 4: 96.2% accurate, 2.0× speedup

Math

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

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(Float64(4.0 / f) / pi)))) / pi))
end

Wolfram

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

Derivation

1. Initial program 6.2%

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

2. Taylor expanded in f around inf 6.2%

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

4. Simplified 6.2%

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

5. Taylor expanded in f around 0 95.7%

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

7. Simplified 95.7%

   \[\leadsto -4 \cdot \frac{\log \color{blue}{\left(\mathsf{fma}\left(f, 0.125 \cdot \frac{{\pi}^{2}}{\pi} - \frac{0.010416666666666666 \cdot {\pi}^{3}}{0.25 \cdot {\pi}^{2}}, \frac{4}{\pi \cdot f}\right) + 0\right)}}{\pi} \]
8. Step-by-step derivation

   1. sub-neg 95.7%

      \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \color{blue}{0.125 \cdot \frac{{\pi}^{2}}{\pi} + \left(-\frac{0.010416666666666666 \cdot {\pi}^{3}}{0.25 \cdot {\pi}^{2}}\right)}, \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

   2. times-frac 95.7%

      \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, 0.125 \cdot \frac{{\pi}^{2}}{\pi} + \left(-\color{blue}{\frac{0.010416666666666666}{0.25} \cdot \frac{{\pi}^{3}}{{\pi}^{2}}}\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

   3. metadata-eval 95.7%

      \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, 0.125 \cdot \frac{{\pi}^{2}}{\pi} + \left(-\color{blue}{0.041666666666666664} \cdot \frac{{\pi}^{3}}{{\pi}^{2}}\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

9. Applied egg-rr 95.7%

   \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \color{blue}{0.125 \cdot \frac{{\pi}^{2}}{\pi} + \left(-0.041666666666666664 \cdot \frac{{\pi}^{3}}{{\pi}^{2}}\right)}, \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]
10. Step-by-step derivation

    1. *-commutative 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\frac{{\pi}^{2}}{\pi} \cdot 0.125} + \left(-0.041666666666666664 \cdot \frac{{\pi}^{3}}{{\pi}^{2}}\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    2. fma-udef 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{fma}\left(\frac{{\pi}^{2}}{\pi}, 0.125, -0.041666666666666664 \cdot \frac{{\pi}^{3}}{{\pi}^{2}}\right)}, \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    3. *-commutative 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{{\pi}^{2}}{\pi}, 0.125, -\color{blue}{\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.041666666666666664}\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    4. unpow2 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{\color{blue}{\pi \cdot \pi}}{\pi}, 0.125, -\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.041666666666666664\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    5. associate-/l* 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\color{blue}{\frac{\pi}{\frac{\pi}{\pi}}}, 0.125, -\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.041666666666666664\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    6. *-inverses 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{\pi}{\color{blue}{1}}, 0.125, -\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.041666666666666664\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    7. /-rgt-identity 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\color{blue}{\pi}, 0.125, -\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.041666666666666664\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    8. distribute-rgt-neg-in 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \color{blue}{\frac{{\pi}^{3}}{{\pi}^{2}} \cdot \left(-0.041666666666666664\right)}\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    9. cube-mult 95.7%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \frac{\color{blue}{\pi \cdot \left(\pi \cdot \pi\right)}}{{\pi}^{2}} \cdot \left(-0.041666666666666664\right)\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    10. unpow2 95.7%

        \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \frac{\pi \cdot \color{blue}{{\pi}^{2}}}{{\pi}^{2}} \cdot \left(-0.041666666666666664\right)\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    11. associate-/l* 95.7%

        \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \color{blue}{\frac{\pi}{\frac{{\pi}^{2}}{{\pi}^{2}}}} \cdot \left(-0.041666666666666664\right)\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    12. *-inverses 95.7%

        \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \frac{\pi}{\color{blue}{1}} \cdot \left(-0.041666666666666664\right)\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    13. /-rgt-identity 95.7%

        \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \color{blue}{\pi} \cdot \left(-0.041666666666666664\right)\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

    14. metadata-eval 95.7%

        \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \pi \cdot \color{blue}{-0.041666666666666664}\right), \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]

11. Simplified 95.7%

    \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{fma}\left(\pi, 0.125, \pi \cdot -0.041666666666666664\right)}, \frac{4}{\pi \cdot f}\right) + 0\right)}{\pi} \]
12. Step-by-step derivation

    1. div-inv 95.5%

       \[\leadsto -4 \cdot \color{blue}{\left(\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, 0.125, \pi \cdot -0.041666666666666664\right), \frac{4}{\pi \cdot f}\right) + 0\right) \cdot \frac{1}{\pi}\right)} \]

    2. +-rgt-identity 95.5%

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

    3. *-commutative 95.5%

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

13. Applied egg-rr 95.5%

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

    1. associate-*r/ 95.7%

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

    2. *-rgt-identity 95.7%

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

    3. fma-def 95.7%

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

    4. distribute-lft-out 95.7%

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

    5. metadata-eval 95.7%

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

    6. associate-/r* 95.7%

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

15. Simplified 95.7%

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

16. Final simplification 95.7%

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

Alternative 5: 95.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.2%

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

2. Taylor expanded in f around 0 95.0%

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

   1. associate-/r* 95.0%

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

   2. distribute-rgt-out-- 95.0%

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

   3. metadata-eval 95.0%

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

4. Simplified 95.0%

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

5. Taylor expanded in f around 0 95.2%

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

   1. neg-mul-1 95.2%

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

   2. sub-neg 95.2%

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

7. Simplified 95.2%

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

8. Final simplification 95.2%

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

Alternative 6: 95.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 1 - \left(1 + \frac{\log \left(2 \cdot \frac{2}{f \cdot \pi}\right)}{0.25 \cdot \pi}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (- 1.0 (+ 1.0 (/ (log (* 2.0 (/ 2.0 (* f PI)))) (* 0.25 PI)))))

C

double code(double f) {
	return 1.0 - (1.0 + (log((2.0 * (2.0 / (f * ((double) M_PI))))) / (0.25 * ((double) M_PI))));
}

Java

public static double code(double f) {
	return 1.0 - (1.0 + (Math.log((2.0 * (2.0 / (f * Math.PI)))) / (0.25 * Math.PI)));
}

Python

def code(f):
	return 1.0 - (1.0 + (math.log((2.0 * (2.0 / (f * math.pi)))) / (0.25 * math.pi)))

Julia

function code(f)
	return Float64(1.0 - Float64(1.0 + Float64(log(Float64(2.0 * Float64(2.0 / Float64(f * pi)))) / Float64(0.25 * pi))))
end

MATLAB

function tmp = code(f)
	tmp = 1.0 - (1.0 + (log((2.0 * (2.0 / (f * pi)))) / (0.25 * pi)));
end

Wolfram

code[f_] := N[(1.0 - N[(1.0 + N[(N[Log[N[(2.0 * N[(2.0 / N[(f * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.25 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.2%

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

2. Taylor expanded in f around 0 95.0%

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

   1. associate-/r* 95.0%

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

   2. distribute-rgt-out-- 95.0%

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

   3. metadata-eval 95.0%

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

4. Simplified 95.0%

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

   1. expm1-log1p-u 93.8%

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

   2. associate-*l/ 93.8%

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

   3. *-un-lft-identity 93.8%

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

   4. associate-/r* 93.8%

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

   5. div-inv 93.8%

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

   6. metadata-eval 93.8%

      \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{\frac{2}{f}}{\pi}}{0.5}\right)}{\pi \cdot \color{blue}{0.25}}\right)\right) \]

6. Applied egg-rr 93.8%

   \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{\frac{2}{f}}{\pi}}{0.5}\right)}{\pi \cdot 0.25}\right)\right)} \]
7. Step-by-step derivation

   1. expm1-udef 93.8%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{\frac{2}{f}}{\pi}}{0.5}\right)}{\pi \cdot 0.25}\right)} - 1\right)} \]

   2. log1p-udef 93.8%

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

   3. add-exp-log 95.1%

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

   4. div-inv 95.1%

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

   5. associate-/l/ 95.1%

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

   6. metadata-eval 95.1%

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

8. Applied egg-rr 95.1%

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

9. Final simplification 95.1%

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

Alternative 7: 95.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.2%

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

2. Taylor expanded in f around 0 95.0%

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

   1. associate-/r* 95.0%

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

   2. distribute-rgt-out-- 95.0%

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

   3. metadata-eval 95.0%

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

4. Simplified 95.0%

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

5. Taylor expanded in f around 0 95.2%

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

   1. neg-mul-1 95.2%

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

   2. sub-neg 95.2%

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

   3. div-sub 95.1%

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

   4. remove-double-neg 95.1%

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

   5. neg-mul-1 95.1%

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

   6. distribute-rgt-neg-in 95.1%

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

   7. log-rec 95.1%

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

   8. div-sub 95.2%

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

7. Simplified 95.1%

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

8. Final simplification 95.1%

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

Alternative 8: 13.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \log 1.3333333333333333 \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (log 1.3333333333333333) (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(1.3333333333333333) * (-1.0 / (((double) M_PI) / 4.0));
}

Java

public static double code(double f) {
	return Math.log(1.3333333333333333) * (-1.0 / (Math.PI / 4.0));
}

Python

def code(f):
	return math.log(1.3333333333333333) * (-1.0 / (math.pi / 4.0))

Julia

function code(f)
	return Float64(log(1.3333333333333333) * Float64(-1.0 / Float64(pi / 4.0)))
end

MATLAB

function tmp = code(f)
	tmp = log(1.3333333333333333) * (-1.0 / (pi / 4.0));
end

Wolfram

code[f_] := N[(N[Log[1.3333333333333333], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.2%

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

3. Applied egg-rr 13.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{1.5}}\right) \]

4. Taylor expanded in f around 0 13.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\log 1.3333333333333333} \]

5. Final simplification 13.3%

   \[\leadsto \log 1.3333333333333333 \cdot \frac{-1}{\frac{\pi}{4}} \]

Alternative 9: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi} \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ (log 7.62939453125e-6) PI) (- 4.0)))

C

double code(double f) {
	return (log(7.62939453125e-6) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return (Math.log(7.62939453125e-6) / Math.PI) * -4.0;
}

Python

def code(f):
	return (math.log(7.62939453125e-6) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(log(7.62939453125e-6) / pi) * Float64(-4.0))
end

MATLAB

function tmp = code(f)
	tmp = (log(7.62939453125e-6) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[Log[7.62939453125e-6], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision]

Derivation

1. Initial program 6.2%

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

3. Applied egg-rr 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{262144}}\right) \]

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}} \]

5. Final simplification 1.6%

   \[\leadsto \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi} \cdot \left(-4\right) \]

Reproduce

herbie shell --seed 2023293 
(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)))))))))