VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.0% → 96.1%

Time: 23.2s

Alternatives: 8

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: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{\pi}{0.5}, -2 \cdot \left(4 \cdot \frac{\pi}{192}\right)\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \cdot \frac{-4}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (fma
    f
    (fma 0.0625 (/ PI 0.5) (* -2.0 (* 4.0 (/ PI 192.0))))
    (/ (/ (/ 2.0 PI) 0.5) f)))
  (/ -4.0 PI)))

C

double code(double f) {
	return log(fma(f, fma(0.0625, (((double) M_PI) / 0.5), (-2.0 * (4.0 * (((double) M_PI) / 192.0)))), (((2.0 / ((double) M_PI)) / 0.5) / f))) * (-4.0 / ((double) M_PI));
}

Julia

function code(f)
	return Float64(log(fma(f, fma(0.0625, Float64(pi / 0.5), Float64(-2.0 * Float64(4.0 * Float64(pi / 192.0)))), Float64(Float64(Float64(2.0 / pi) / 0.5) / f))) * Float64(-4.0 / pi))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(0.0625 * N[(Pi / 0.5), $MachinePrecision] + N[(-2.0 * N[(4.0 * N[(Pi / 192.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 / Pi), $MachinePrecision] / 0.5), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

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

   1. distribute-lft-neg-in 6.6%

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

   2. *-commutative 6.6%

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

3. Simplified 6.6%

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

4. Taylor expanded in f around 0 96.1%

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

6. Simplified 96.1%

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

   1. fma-udef 96.1%

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

   2. associate-/r* 96.1%

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

   3. pow1 96.1%

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

   4. pow-div 96.1%

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

   5. metadata-eval 96.1%

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

   6. pow1 96.1%

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

   7. associate-*l/ 96.1%

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

   8. div-inv 96.1%

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

   9. unpow-prod-down 96.1%

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

   10. metadata-eval 96.1%

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

   11. metadata-eval 96.1%

       \[\leadsto \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \frac{\pi}{0.5} + \frac{{\pi}^{3} \cdot -2}{\left({\pi}^{2} \cdot 0.25\right) \cdot \color{blue}{192}}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

8. Applied egg-rr 96.1%

   \[\leadsto \log \left(\mathsf{fma}\left(f, \color{blue}{0.0625 \cdot \frac{\pi}{0.5} + \frac{{\pi}^{3} \cdot -2}{\left({\pi}^{2} \cdot 0.25\right) \cdot 192}}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]
9. Step-by-step derivation

   1. fma-def 96.1%

      \[\leadsto \log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{fma}\left(0.0625, \frac{\pi}{0.5}, \frac{{\pi}^{3} \cdot -2}{\left({\pi}^{2} \cdot 0.25\right) \cdot 192}\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

   2. *-commutative 96.1%

      \[\leadsto \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{\pi}{0.5}, \frac{\color{blue}{-2 \cdot {\pi}^{3}}}{\left({\pi}^{2} \cdot 0.25\right) \cdot 192}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

   3. *-lft-identity 96.1%

      \[\leadsto \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{\pi}{0.5}, \frac{-2 \cdot {\pi}^{3}}{\color{blue}{1 \cdot \left(\left({\pi}^{2} \cdot 0.25\right) \cdot 192\right)}}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

   4. times-frac 96.1%

      \[\leadsto \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{\pi}{0.5}, \color{blue}{\frac{-2}{1} \cdot \frac{{\pi}^{3}}{\left({\pi}^{2} \cdot 0.25\right) \cdot 192}}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

   5. metadata-eval 96.1%

      \[\leadsto \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{\pi}{0.5}, \color{blue}{-2} \cdot \frac{{\pi}^{3}}{\left({\pi}^{2} \cdot 0.25\right) \cdot 192}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

   6. metadata-eval 96.1%

      \[\leadsto \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{\pi}{0.5}, -2 \cdot \frac{{\pi}^{\color{blue}{\left(2 + 1\right)}}}{\left({\pi}^{2} \cdot 0.25\right) \cdot 192}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

   7. pow-plus 96.1%

      \[\leadsto \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{\pi}{0.5}, -2 \cdot \frac{\color{blue}{{\pi}^{2} \cdot \pi}}{\left({\pi}^{2} \cdot 0.25\right) \cdot 192}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

   8. times-frac 96.1%

      \[\leadsto \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{\pi}{0.5}, -2 \cdot \color{blue}{\left(\frac{{\pi}^{2}}{{\pi}^{2} \cdot 0.25} \cdot \frac{\pi}{192}\right)}\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

10. Simplified 96.1%

    \[\leadsto \log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{fma}\left(0.0625, \frac{\pi}{0.5}, -2 \cdot \left(4 \cdot \frac{\pi}{192}\right)\right)}, \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right) + 0\right) \cdot \frac{-4}{\pi} \]

11. Final simplification 96.1%

    \[\leadsto \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, \frac{\pi}{0.5}, -2 \cdot \left(4 \cdot \frac{\pi}{192}\right)\right), \frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)\right) \cdot \frac{-4}{\pi} \]

Alternative 2: 95.8% 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 6.6%

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

   1. distribute-lft-neg-in 6.6%

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

   2. *-commutative 6.6%

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

3. Simplified 6.6%

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

4. Taylor expanded in f around 0 95.9%

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

   1. distribute-rgt-out-- 95.9%

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

   2. metadata-eval 95.9%

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

6. Simplified 95.9%

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

7. Taylor expanded in f around 0 96.0%

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

8. Final simplification 96.0%

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

Alternative 3: 95.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

   1. distribute-lft-neg-in 6.6%

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

   2. *-commutative 6.6%

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

3. Simplified 6.6%

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

4. Taylor expanded in f around 0 95.9%

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

   1. distribute-rgt-out-- 95.9%

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

   2. metadata-eval 95.9%

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

6. Simplified 95.9%

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

7. Taylor expanded in f around 0 95.9%

   \[\leadsto \log \color{blue}{\left(0.125 \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)} \cdot \frac{-4}{\pi} \]
8. Step-by-step derivation

   1. associate-*r/ 96.0%

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

   2. fma-def 96.0%

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

   3. *-commutative 96.0%

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

   4. un-div-inv 96.0%

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

   5. associate-/l/ 96.0%

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

9. Applied egg-rr 96.0%

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

10. Final simplification 96.0%

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

Alternative 4: 95.7% 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 6.6%

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

   1. distribute-lft-neg-in 6.6%

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

   2. *-commutative 6.6%

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

3. Simplified 6.6%

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

4. Taylor expanded in f around 0 95.9%

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

   1. mul-1-neg 95.9%

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

   2. unsub-neg 95.9%

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

   3. distribute-rgt-out-- 95.9%

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

   4. metadata-eval 95.9%

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

   5. metadata-eval 95.9%

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

   6. associate-/r* 95.9%

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

   7. metadata-eval 95.9%

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

6. Simplified 95.9%

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

7. Final simplification 95.9%

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

Alternative 5: 1.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

   1. distribute-lft-neg-in 6.6%

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

   2. *-commutative 6.6%

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

3. Simplified 6.6%

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

4. Taylor expanded in f around 0 95.9%

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

   1. distribute-rgt-out-- 95.9%

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

   2. metadata-eval 95.9%

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

6. Simplified 95.9%

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

7. Taylor expanded in f around 0 95.9%

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

8. Taylor expanded in f around inf 1.6%

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

   1. associate-*r* 1.6%

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

   2. *-commutative 1.6%

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

   3. *-commutative 1.6%

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

10. Simplified 1.6%

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

11. Final simplification 1.6%

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

Alternative 6: 95.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

   1. distribute-lft-neg-in 6.6%

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

   2. *-commutative 6.6%

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

3. Simplified 6.6%

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

4. Taylor expanded in f around 0 95.8%

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

   1. *-commutative 95.8%

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

   2. associate-/r* 95.8%

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

   3. distribute-rgt-out-- 95.8%

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

   4. metadata-eval 95.8%

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

   5. metadata-eval 95.8%

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

   6. associate-/r* 95.8%

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

   7. metadata-eval 95.8%

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

6. Simplified 95.8%

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

   1. div-inv 95.8%

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

   2. div-inv 95.8%

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

   3. metadata-eval 95.8%

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

8. Applied egg-rr 95.8%

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

   1. associate-*r/ 95.8%

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

   2. *-rgt-identity 95.8%

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

   3. associate-*l/ 95.8%

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

   4. metadata-eval 95.8%

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

10. Simplified 95.8%

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

11. Final simplification 95.8%

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

Alternative 7: 95.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

   1. distribute-lft-neg-in 6.6%

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

   2. *-commutative 6.6%

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

3. Simplified 6.6%

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

4. Taylor expanded in f around 0 95.8%

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

   1. *-commutative 95.8%

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

   2. associate-/r* 95.8%

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

   3. distribute-rgt-out-- 95.8%

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

   4. metadata-eval 95.8%

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

   5. metadata-eval 95.8%

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

   6. associate-/r* 95.8%

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

   7. metadata-eval 95.8%

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

6. Simplified 95.8%

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

   1. div-inv 95.8%

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

   2. div-inv 95.8%

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

   3. metadata-eval 95.8%

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

8. Applied egg-rr 95.8%

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

   1. associate-*r/ 95.8%

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

   2. *-rgt-identity 95.8%

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

   3. associate-*l/ 95.8%

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

   4. metadata-eval 95.8%

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

10. Simplified 95.8%

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

    1. associate-*r/ 95.9%

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

12. Applied egg-rr 95.9%

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

13. Final simplification 95.9%

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

Alternative 8: 0.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

   1. distribute-lft-neg-in 6.6%

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

   2. *-commutative 6.6%

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

3. Simplified 6.6%

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

4. Taylor expanded in f around 0 95.8%

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

5. Taylor expanded in f around inf 0.7%

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

   1. distribute-rgt-out 0.7%

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

   2. distribute-rgt-out-- 0.7%

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

   3. metadata-eval 0.7%

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

   4. metadata-eval 0.7%

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

   5. mul0-rgt 0.7%

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

7. Simplified 0.7%

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

8. Final simplification 0.7%

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

Reproduce

herbie shell --seed 2023326 
(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)))))))))