VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 96.5%

Time: 21.6s

Alternatives: 10

Speedup: 3.3×

• 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: 6.7% 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: 96.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \frac{\pi}{0.5} + -2 \cdot \left(\frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot 0.005208333333333333\right), 0 \cdot {\left(\pi \cdot 0.5\right)}^{2}\right)\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (fma
  -4.0
  (/ (- (log (/ (/ 2.0 PI) 0.5)) (log f)) PI)
  (*
   -2.0
   (+
    (* (/ f PI) (* PI 0.0))
    (*
     (/ (pow f 2.0) PI)
     (fma
      (* PI 0.5)
      (+
       (* 0.0625 (/ PI 0.5))
       (*
        -2.0
        (* (/ (pow PI 3.0) (* 0.25 (pow PI 2.0))) 0.005208333333333333)))
      (* 0.0 (pow (* PI 0.5) 2.0))))))))

C

double code(double f) {
	return fma(-4.0, ((log(((2.0 / ((double) M_PI)) / 0.5)) - log(f)) / ((double) M_PI)), (-2.0 * (((f / ((double) M_PI)) * (((double) M_PI) * 0.0)) + ((pow(f, 2.0) / ((double) M_PI)) * fma((((double) M_PI) * 0.5), ((0.0625 * (((double) M_PI) / 0.5)) + (-2.0 * ((pow(((double) M_PI), 3.0) / (0.25 * pow(((double) M_PI), 2.0))) * 0.005208333333333333))), (0.0 * pow((((double) M_PI) * 0.5), 2.0)))))));
}

Julia

function code(f)
	return fma(-4.0, Float64(Float64(log(Float64(Float64(2.0 / pi) / 0.5)) - log(f)) / pi), Float64(-2.0 * Float64(Float64(Float64(f / pi) * Float64(pi * 0.0)) + Float64(Float64((f ^ 2.0) / pi) * fma(Float64(pi * 0.5), Float64(Float64(0.0625 * Float64(pi / 0.5)) + Float64(-2.0 * Float64(Float64((pi ^ 3.0) / Float64(0.25 * (pi ^ 2.0))) * 0.005208333333333333))), Float64(0.0 * (Float64(pi * 0.5) ^ 2.0)))))))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[(N[Log[N[(N[(2.0 / Pi), $MachinePrecision] / 0.5), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] + N[(-2.0 * N[(N[(N[(f / Pi), $MachinePrecision] * N[(Pi * 0.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[f, 2.0], $MachinePrecision] / Pi), $MachinePrecision] * N[(N[(Pi * 0.5), $MachinePrecision] * N[(N[(0.0625 * N[(Pi / 0.5), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(N[Power[Pi, 3.0], $MachinePrecision] / N[(0.25 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.005208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0 * N[Power[N[(Pi * 0.5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.3%

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

   1. distribute-lft-neg-in 8.3%

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

   2. *-commutative 8.3%

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

3. Simplified 8.3%

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

4. Taylor expanded in f around 0 95.9%

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

6. Simplified 95.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2\right), {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right)} \]
7. Step-by-step derivation

   1. fma-udef 95.9%

      \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{0.0625 \cdot \frac{{\pi}^{2}}{\pi \cdot 0.5} + \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2}, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

   2. associate-/r* 95.9%

      \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \color{blue}{\frac{\frac{{\pi}^{2}}{\pi}}{0.5}} + \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

   3. pow1 95.9%

      \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \frac{\frac{{\pi}^{2}}{\color{blue}{{\pi}^{1}}}}{0.5} + \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

   4. pow-div 95.9%

      \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \frac{\color{blue}{{\pi}^{\left(2 - 1\right)}}}{0.5} + \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

   5. metadata-eval 95.9%

      \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \frac{{\pi}^{\color{blue}{1}}}{0.5} + \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

   6. pow1 95.9%

      \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \frac{\color{blue}{\pi}}{0.5} + \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

   7. associate-/r/ 95.9%

      \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \frac{\pi}{0.5} + \color{blue}{\left(\frac{{\pi}^{3}}{{\left(\pi \cdot 0.5\right)}^{2}} \cdot 0.005208333333333333\right)} \cdot -2, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

   8. *-commutative 95.9%

      \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \frac{\pi}{0.5} + \left(\frac{{\pi}^{3}}{{\color{blue}{\left(0.5 \cdot \pi\right)}}^{2}} \cdot 0.005208333333333333\right) \cdot -2, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

   9. unpow-prod-down 95.9%

      \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \frac{\pi}{0.5} + \left(\frac{{\pi}^{3}}{\color{blue}{{0.5}^{2} \cdot {\pi}^{2}}} \cdot 0.005208333333333333\right) \cdot -2, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

   10. metadata-eval 95.9%

       \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \frac{\pi}{0.5} + \left(\frac{{\pi}^{3}}{\color{blue}{0.25} \cdot {\pi}^{2}} \cdot 0.005208333333333333\right) \cdot -2, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

8. Applied egg-rr 95.9%

   \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{0.0625 \cdot \frac{\pi}{0.5} + \left(\frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot 0.005208333333333333\right) \cdot -2}, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

9. Final simplification 95.9%

   \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \frac{\pi}{0.5} + -2 \cdot \left(\frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot 0.005208333333333333\right), 0 \cdot {\left(\pi \cdot 0.5\right)}^{2}\right)\right)\right) \]

Alternative 2: 96.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, -2 \cdot \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}}\right), {\left(\sqrt{\frac{\frac{2}{\pi}}{0.5 \cdot f}}\right)}^{2}\right)\right) \cdot \frac{-4}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (fma
    f
    (fma
     0.0625
     (/ (pow PI 2.0) (* PI 0.5))
     (* -2.0 (/ (pow PI 3.0) (/ (pow (* PI 0.5) 2.0) 0.005208333333333333))))
    (pow (sqrt (/ (/ 2.0 PI) (* 0.5 f))) 2.0)))
  (/ -4.0 PI)))

C

double code(double f) {
	return log(fma(f, fma(0.0625, (pow(((double) M_PI), 2.0) / (((double) M_PI) * 0.5)), (-2.0 * (pow(((double) M_PI), 3.0) / (pow((((double) M_PI) * 0.5), 2.0) / 0.005208333333333333)))), pow(sqrt(((2.0 / ((double) M_PI)) / (0.5 * f))), 2.0))) * (-4.0 / ((double) M_PI));
}

Julia

function code(f)
	return Float64(log(fma(f, fma(0.0625, Float64((pi ^ 2.0) / Float64(pi * 0.5)), Float64(-2.0 * Float64((pi ^ 3.0) / Float64((Float64(pi * 0.5) ^ 2.0) / 0.005208333333333333)))), (sqrt(Float64(Float64(2.0 / pi) / Float64(0.5 * f))) ^ 2.0))) * Float64(-4.0 / pi))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(0.0625 * N[(N[Power[Pi, 2.0], $MachinePrecision] / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[Pi, 3.0], $MachinePrecision] / N[(N[Power[N[(Pi * 0.5), $MachinePrecision], 2.0], $MachinePrecision] / 0.005208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sqrt[N[(N[(2.0 / Pi), $MachinePrecision] / N[(0.5 * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.3%

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

   1. distribute-lft-neg-in 8.3%

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

   2. *-commutative 8.3%

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

3. Simplified 8.3%

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

4. Taylor expanded in f around 0 95.7%

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

6. Simplified 95.7%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right)} \cdot \frac{-4}{\pi} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 95.7%

      \[\leadsto \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2\right), \color{blue}{\sqrt{\frac{\frac{\frac{2}{\pi}}{0.5}}{f}} \cdot \sqrt{\frac{\frac{\frac{2}{\pi}}{0.5}}{f}}}\right) + 0\right) \cdot \frac{-4}{\pi} \]

   2. pow2 95.7%

      \[\leadsto \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2\right), \color{blue}{{\left(\sqrt{\frac{\frac{\frac{2}{\pi}}{0.5}}{f}}\right)}^{2}}\right) + 0\right) \cdot \frac{-4}{\pi} \]

   3. associate-/l/ 95.7%

      \[\leadsto \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2\right), {\left(\sqrt{\color{blue}{\frac{\frac{2}{\pi}}{f \cdot 0.5}}}\right)}^{2}\right) + 0\right) \cdot \frac{-4}{\pi} \]

8. Applied egg-rr 95.7%

   \[\leadsto \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2\right), \color{blue}{{\left(\sqrt{\frac{\frac{2}{\pi}}{f \cdot 0.5}}\right)}^{2}}\right) + 0\right) \cdot \frac{-4}{\pi} \]

9. Final simplification 95.7%

   \[\leadsto \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, -2 \cdot \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}}\right), {\left(\sqrt{\frac{\frac{2}{\pi}}{0.5 \cdot f}}\right)}^{2}\right)\right) \cdot \frac{-4}{\pi} \]

Alternative 3: 96.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{-4}{\pi} \cdot \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + -2 \cdot \left(\pi \cdot 0.020833333333333332\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ -4.0 PI)
  (log
   (fma
    f
    (+ (* 0.0625 (/ PI 0.5)) (* -2.0 (* PI 0.020833333333333332)))
    (/ (/ (/ 2.0 PI) 0.5) f)))))

C

double code(double f) {
	return (-4.0 / ((double) M_PI)) * log(fma(f, ((0.0625 * (((double) M_PI) / 0.5)) + (-2.0 * (((double) M_PI) * 0.020833333333333332))), (((2.0 / ((double) M_PI)) / 0.5) / f)));
}

Julia

function code(f)
	return Float64(Float64(-4.0 / pi) * log(fma(f, Float64(Float64(0.0625 * Float64(pi / 0.5)) + Float64(-2.0 * Float64(pi * 0.020833333333333332))), Float64(Float64(Float64(2.0 / pi) / 0.5) / f))))
end

Wolfram

code[f_] := N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(f * N[(N[(0.0625 * N[(Pi / 0.5), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(Pi * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 / Pi), $MachinePrecision] / 0.5), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.3%

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

   1. distribute-lft-neg-in 8.3%

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

   2. *-commutative 8.3%

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

3. Simplified 8.3%

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

4. Taylor expanded in f around 0 95.7%

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

6. Simplified 95.7%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{{\pi}^{2}}{\pi \cdot 0.5}, \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right)} \cdot \frac{-4}{\pi} \]
7. Step-by-step derivation

   1. fma-udef 95.9%

      \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{0.0625 \cdot \frac{{\pi}^{2}}{\pi \cdot 0.5} + \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2}, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

   2. associate-/r* 95.9%

      \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \color{blue}{\frac{\frac{{\pi}^{2}}{\pi}}{0.5}} + \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

   3. pow1 95.9%

      \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \frac{\frac{{\pi}^{2}}{\color{blue}{{\pi}^{1}}}}{0.5} + \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

   4. pow-div 95.9%

      \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \frac{\color{blue}{{\pi}^{\left(2 - 1\right)}}}{0.5} + \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

   5. metadata-eval 95.9%

      \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \frac{{\pi}^{\color{blue}{1}}}{0.5} + \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

   6. pow1 95.9%

      \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \frac{\color{blue}{\pi}}{0.5} + \frac{{\pi}^{3}}{\frac{{\left(\pi \cdot 0.5\right)}^{2}}{0.005208333333333333}} \cdot -2, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

   7. associate-/r/ 95.9%

      \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \frac{\pi}{0.5} + \color{blue}{\left(\frac{{\pi}^{3}}{{\left(\pi \cdot 0.5\right)}^{2}} \cdot 0.005208333333333333\right)} \cdot -2, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

   8. *-commutative 95.9%

      \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \frac{\pi}{0.5} + \left(\frac{{\pi}^{3}}{{\color{blue}{\left(0.5 \cdot \pi\right)}}^{2}} \cdot 0.005208333333333333\right) \cdot -2, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

   9. unpow-prod-down 95.9%

      \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \frac{\pi}{0.5} + \left(\frac{{\pi}^{3}}{\color{blue}{{0.5}^{2} \cdot {\pi}^{2}}} \cdot 0.005208333333333333\right) \cdot -2, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

   10. metadata-eval 95.9%

       \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -2 \cdot \left(\frac{f}{\pi} \cdot \left(\pi \cdot 0\right) + \frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi \cdot 0.5, 0.0625 \cdot \frac{\pi}{0.5} + \left(\frac{{\pi}^{3}}{\color{blue}{0.25} \cdot {\pi}^{2}} \cdot 0.005208333333333333\right) \cdot -2, {\left(\pi \cdot 0.5\right)}^{2} \cdot 0\right)\right)\right) \]

8. Applied egg-rr 95.7%

   \[\leadsto \log \left(\mathsf{fma}\left(f, \color{blue}{0.0625 \cdot \frac{\pi}{0.5} + \left(\frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot 0.005208333333333333\right) \cdot -2}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]
9. Step-by-step derivation

   1. expm1-log1p-u 95.7%

      \[\leadsto \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot 0.005208333333333333\right)\right)} \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

   2. expm1-udef 95.7%

      \[\leadsto \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot 0.005208333333333333\right)} - 1\right)} \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

   3. associate-*l/ 95.7%

      \[\leadsto \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\pi}^{3} \cdot 0.005208333333333333}{0.25 \cdot {\pi}^{2}}}\right)} - 1\right) \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

   4. *-commutative 95.7%

      \[\leadsto \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{3} \cdot 0.005208333333333333}{\color{blue}{{\pi}^{2} \cdot 0.25}}\right)} - 1\right) \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

   5. times-frac 95.7%

      \[\leadsto \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\pi}^{3}}{{\pi}^{2}} \cdot \frac{0.005208333333333333}{0.25}}\right)} - 1\right) \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

   6. metadata-eval 95.7%

      \[\leadsto \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + \left(e^{\mathsf{log1p}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot \color{blue}{0.020833333333333332}\right)} - 1\right) \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

10. Applied egg-rr 95.7%

    \[\leadsto \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right)} - 1\right)} \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]
11. Step-by-step derivation

    1. expm1-def 95.7%

       \[\leadsto \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right)\right)} \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    2. expm1-log1p 95.7%

       \[\leadsto \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + \color{blue}{\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right)} \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    3. cube-mult 95.7%

       \[\leadsto \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + \left(\frac{\color{blue}{\pi \cdot \left(\pi \cdot \pi\right)}}{{\pi}^{2}} \cdot 0.020833333333333332\right) \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    4. unpow2 95.7%

       \[\leadsto \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + \left(\frac{\pi \cdot \color{blue}{{\pi}^{2}}}{{\pi}^{2}} \cdot 0.020833333333333332\right) \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    5. associate-/l* 95.7%

       \[\leadsto \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + \left(\color{blue}{\frac{\pi}{\frac{{\pi}^{2}}{{\pi}^{2}}}} \cdot 0.020833333333333332\right) \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    6. *-inverses 95.7%

       \[\leadsto \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + \left(\frac{\pi}{\color{blue}{1}} \cdot 0.020833333333333332\right) \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

    7. /-rgt-identity 95.7%

       \[\leadsto \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + \left(\color{blue}{\pi} \cdot 0.020833333333333332\right) \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

12. Simplified 95.7%

    \[\leadsto \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + \color{blue}{\left(\pi \cdot 0.020833333333333332\right)} \cdot -2, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

13. Final simplification 95.7%

    \[\leadsto \frac{-4}{\pi} \cdot \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + -2 \cdot \left(\pi \cdot 0.020833333333333332\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \]

Alternative 4: 96.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} + -0.125 \cdot \left(\pi \cdot {f}^{2}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (+
  (* -4.0 (/ (- (log (/ 4.0 PI)) (log f)) PI))
  (* -0.125 (* PI (pow f 2.0)))))

C

double code(double f) {
	return (-4.0 * ((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI))) + (-0.125 * (((double) M_PI) * pow(f, 2.0)));
}

Java

public static double code(double f) {
	return (-4.0 * ((Math.log((4.0 / Math.PI)) - Math.log(f)) / Math.PI)) + (-0.125 * (Math.PI * Math.pow(f, 2.0)));
}

Python

def code(f):
	return (-4.0 * ((math.log((4.0 / math.pi)) - math.log(f)) / math.pi)) + (-0.125 * (math.pi * math.pow(f, 2.0)))

Julia

function code(f)
	return Float64(Float64(-4.0 * Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi)) + Float64(-0.125 * Float64(pi * (f ^ 2.0))))
end

MATLAB

function tmp = code(f)
	tmp = (-4.0 * ((log((4.0 / pi)) - log(f)) / pi)) + (-0.125 * (pi * (f ^ 2.0)));
end

Wolfram

code[f_] := N[(N[(-4.0 * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] + N[(-0.125 * N[(Pi * N[Power[f, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.3%

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

   1. distribute-lft-neg-in 8.3%

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

   2. *-commutative 8.3%

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

3. Simplified 8.3%

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

4. Taylor expanded in f around 0 95.2%

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

   1. distribute-rgt-out-- 95.2%

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

   2. metadata-eval 95.2%

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

6. Simplified 95.2%

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

7. Taylor expanded in f around 0 95.4%

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

8. Final simplification 95.4%

   \[\leadsto -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} + -0.125 \cdot \left(\pi \cdot {f}^{2}\right) \]

Alternative 5: 95.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{-4}{\pi} \cdot \log \left(0.125 \cdot \left(\pi \cdot f\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (/ -4.0 PI) (log (+ (* 0.125 (* PI f)) (* 4.0 (/ 1.0 (* PI f)))))))

C

double code(double f) {
	return (-4.0 / ((double) M_PI)) * log(((0.125 * (((double) M_PI) * f)) + (4.0 * (1.0 / (((double) M_PI) * f)))));
}

Java

public static double code(double f) {
	return (-4.0 / Math.PI) * Math.log(((0.125 * (Math.PI * f)) + (4.0 * (1.0 / (Math.PI * f)))));
}

Python

def code(f):
	return (-4.0 / math.pi) * math.log(((0.125 * (math.pi * f)) + (4.0 * (1.0 / (math.pi * f)))))

Julia

function code(f)
	return Float64(Float64(-4.0 / pi) * log(Float64(Float64(0.125 * Float64(pi * f)) + Float64(4.0 * Float64(1.0 / Float64(pi * f))))))
end

MATLAB

function tmp = code(f)
	tmp = (-4.0 / pi) * log(((0.125 * (pi * f)) + (4.0 * (1.0 / (pi * f)))));
end

Wolfram

code[f_] := N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(N[(0.125 * N[(Pi * f), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(1.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.3%

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

   1. distribute-lft-neg-in 8.3%

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

   2. *-commutative 8.3%

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

3. Simplified 8.3%

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

4. Taylor expanded in f around 0 95.2%

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

   1. distribute-rgt-out-- 95.2%

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

   2. metadata-eval 95.2%

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

6. Simplified 95.2%

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

7. Taylor expanded in f around 0 95.2%

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

8. Final simplification 95.2%

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

Alternative 6: 96.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

   1. distribute-lft-neg-in 8.3%

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

   2. *-commutative 8.3%

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

3. Simplified 8.3%

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

4. Taylor expanded in f around 0 95.2%

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

   1. mul-1-neg 95.2%

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

   2. unsub-neg 95.2%

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

   3. distribute-rgt-out-- 95.2%

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

   4. metadata-eval 95.2%

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

   5. metadata-eval 95.2%

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

   6. associate-/r* 95.2%

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

   7. metadata-eval 95.2%

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

6. Simplified 95.2%

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

7. Final simplification 95.2%

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

Alternative 7: 95.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

   1. distribute-lft-neg-in 8.3%

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

   2. *-commutative 8.3%

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

3. Simplified 8.3%

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

4. Taylor expanded in f around 0 95.0%

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

   1. *-commutative 95.0%

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

   2. associate-/r* 95.0%

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

   3. distribute-rgt-out-- 95.0%

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

   4. metadata-eval 95.0%

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

   5. metadata-eval 95.0%

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

   6. associate-/r* 95.0%

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

   7. metadata-eval 95.0%

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

6. Simplified 95.0%

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

7. Taylor expanded in f around 0 95.2%

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

9. Simplified 95.0%

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

    1. add-sqr-sqrt 94.5%

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

    2. sqrt-unprod 95.2%

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

    3. pow2 95.2%

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

    4. *-commutative 95.2%

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

11. Applied egg-rr 95.2%

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

    1. unpow2 95.2%

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

    2. rem-sqrt-square 95.2%

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

13. Simplified 95.2%

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

14. Final simplification 95.2%

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

Alternative 8: 95.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

   1. distribute-lft-neg-in 8.3%

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

   2. *-commutative 8.3%

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

3. Simplified 8.3%

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

4. Taylor expanded in f around 0 95.0%

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

   1. *-commutative 95.0%

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

   2. associate-/r* 95.0%

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

   3. distribute-rgt-out-- 95.0%

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

   4. metadata-eval 95.0%

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

   5. metadata-eval 95.0%

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

   6. associate-/r* 95.0%

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

   7. metadata-eval 95.0%

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

6. Simplified 95.0%

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

7. Taylor expanded in f around 0 95.2%

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

9. Simplified 95.0%

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

10. Final simplification 95.0%

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

Alternative 9: 95.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

   1. distribute-lft-neg-in 8.3%

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

   2. *-commutative 8.3%

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

3. Simplified 8.3%

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

4. Taylor expanded in f around 0 95.0%

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

   1. *-commutative 95.0%

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

   2. associate-/r* 95.0%

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

   3. distribute-rgt-out-- 95.0%

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

   4. metadata-eval 95.0%

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

   5. metadata-eval 95.0%

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

   6. associate-/r* 95.0%

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

   7. metadata-eval 95.0%

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

6. Simplified 95.0%

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

7. Taylor expanded in f around 0 95.2%

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

9. Simplified 95.0%

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

    1. *-commutative 95.0%

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

    2. associate-*r/ 95.2%

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

    3. *-commutative 95.2%

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

11. Applied egg-rr 95.2%

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

12. Final simplification 95.2%

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

Alternative 10: 0.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{-4 \cdot \log 0}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (* -4.0 (log 0.0)) PI))

C

double code(double f) {
	return (-4.0 * log(0.0)) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (-4.0 * Math.log(0.0)) / Math.PI;
}

Python

def code(f):
	return (-4.0 * math.log(0.0)) / math.pi

Julia

function code(f)
	return Float64(Float64(-4.0 * log(0.0)) / pi)
end

MATLAB

function tmp = code(f)
	tmp = (-4.0 * log(0.0)) / pi;
end

Wolfram

code[f_] := N[(N[(-4.0 * N[Log[0.0], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 8.3%

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

   1. distribute-lft-neg-in 8.3%

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

   2. *-commutative 8.3%

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

3. Simplified 8.3%

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

4. Taylor expanded in f around 0 95.0%

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

5. Taylor expanded in f around inf 0.7%

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

   1. associate-*r/ 0.7%

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

   2. distribute-rgt-out 0.7%

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

   3. distribute-rgt-out-- 0.7%

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

   4. metadata-eval 0.7%

      \[\leadsto \frac{-4 \cdot \log \left(\frac{\pi}{\pi \cdot \color{blue}{0.5}} \cdot \left(-0.25 + 0.25\right)\right)}{\pi} \]

   5. metadata-eval 0.7%

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

   6. mul0-rgt 0.7%

      \[\leadsto \frac{-4 \cdot \log \color{blue}{0}}{\pi} \]

7. Simplified 0.7%

   \[\leadsto \color{blue}{\frac{-4 \cdot \log 0}{\pi}} \]

8. Final simplification 0.7%

   \[\leadsto \frac{-4 \cdot \log 0}{\pi} \]

Reproduce

herbie shell --seed 2023332 
(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)))))))))