VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 96.5%

Time: 25.5s

Alternatives: 4

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: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ -\mathsf{fma}\left(4, \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}, {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (fma
   4.0
   (/ (log (/ (/ 4.0 PI) f)) PI)
   (* (pow f 2.0) (* PI 0.08333333333333333)))))

C

double code(double f) {
	return -fma(4.0, (log(((4.0 / ((double) M_PI)) / f)) / ((double) M_PI)), (pow(f, 2.0) * (((double) M_PI) * 0.08333333333333333)));
}

Julia

function code(f)
	return Float64(-fma(4.0, Float64(log(Float64(Float64(4.0 / pi) / f)) / pi), Float64((f ^ 2.0) * Float64(pi * 0.08333333333333333))))
end

Wolfram

code[f_] := (-N[(4.0 * N[(N[Log[N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] + N[(N[Power[f, 2.0], $MachinePrecision] * N[(Pi * 0.08333333333333333), $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. Taylor expanded in f around 0 94.9%

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

   1. +-commutative 94.9%

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

   2. associate-+l+ 94.9%

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

   3. fma-def 94.9%

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

   4. distribute-rgt-out-- 94.9%

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

   5. metadata-eval 94.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}\right) \]

   6. fma-def 94.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \color{blue}{\mathsf{fma}\left({f}^{5}, 8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}, f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}\right)}\right) \]

   7. distribute-rgt-out-- 94.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, \color{blue}{{\pi}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} - -8.138020833333333 \cdot 10^{-6}\right)}, f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right)}\right) \]

   8. metadata-eval 94.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}, f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right)}\right) \]

   9. distribute-rgt-out-- 94.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \color{blue}{\left(\pi \cdot \left(0.25 - -0.25\right)\right)}\right)\right)}\right) \]

4. Simplified 94.9%

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

5. Taylor expanded in f around 0 94.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right)\right)\right)} \]

7. Simplified 94.8%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\frac{\frac{4}{\pi}}{f}\right) + 0.5 \cdot \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right)\right)} \]

8. Taylor expanded in f around 0 95.0%

   \[\leadsto -\color{blue}{\left(0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right) + 4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}{\pi}\right)} \]
9. Step-by-step derivation

   1. +-commutative 95.0%

      \[\leadsto -\color{blue}{\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}{\pi} + 0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right)} \]

   2. mul-1-neg 95.0%

      \[\leadsto -\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(-\log f\right)}}{\pi} + 0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right) \]

   3. log-rec 95.0%

      \[\leadsto -\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + \color{blue}{\log \left(\frac{1}{f}\right)}}{\pi} + 0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right) \]

   4. +-commutative 95.0%

      \[\leadsto -\left(4 \cdot \frac{\color{blue}{\log \left(\frac{1}{f}\right) + \log \left(\frac{4}{\pi}\right)}}{\pi} + 0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right) \]

   5. fma-def 95.0%

      \[\leadsto -\color{blue}{\mathsf{fma}\left(4, \frac{\log \left(\frac{1}{f}\right) + \log \left(\frac{4}{\pi}\right)}{\pi}, 0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right)} \]

   6. +-commutative 95.0%

      \[\leadsto -\mathsf{fma}\left(4, \frac{\color{blue}{\log \left(\frac{4}{\pi}\right) + \log \left(\frac{1}{f}\right)}}{\pi}, 0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right) \]

   7. log-rec 95.0%

      \[\leadsto -\mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(-\log f\right)}}{\pi}, 0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right) \]

   8. sub-neg 95.0%

      \[\leadsto -\mathsf{fma}\left(4, \frac{\color{blue}{\log \left(\frac{4}{\pi}\right) - \log f}}{\pi}, 0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right) \]

   9. log-div 95.0%

      \[\leadsto -\mathsf{fma}\left(4, \frac{\color{blue}{\log \left(\frac{\frac{4}{\pi}}{f}\right)}}{\pi}, 0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right) \]

   10. *-commutative 95.0%

       \[\leadsto -\mathsf{fma}\left(4, \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}, \color{blue}{\left({f}^{2} \cdot \pi\right) \cdot 0.08333333333333333}\right) \]

   11. associate-*l* 95.0%

       \[\leadsto -\mathsf{fma}\left(4, \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}, \color{blue}{{f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right)}\right) \]

10. Simplified 95.0%

    \[\leadsto -\color{blue}{\mathsf{fma}\left(4, \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}, {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right)\right)} \]

11. Final simplification 95.0%

    \[\leadsto -\mathsf{fma}\left(4, \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}, {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right)\right) \]

Alternative 2: 96.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(4.0 * (-N[Abs[N[(N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $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. Taylor expanded in f around 0 94.4%

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

   1. associate-/r* 94.4%

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

   2. distribute-rgt-out-- 94.4%

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

   3. metadata-eval 94.4%

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

4. Simplified 94.4%

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

5. Taylor expanded in f around 0 94.5%

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

   1. neg-mul-1 94.5%

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

   2. sub-neg 94.5%

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

   3. metadata-eval 94.5%

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

   4. associate-*l/ 94.5%

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

   5. log-div 94.5%

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

   6. associate-*l/ 94.5%

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

   7. metadata-eval 94.5%

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

7. Simplified 94.5%

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

   1. add-sqr-sqrt 94.0%

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

   2. sqrt-unprod 94.6%

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

   3. pow2 94.6%

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

   4. associate-/l/ 94.6%

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

   5. *-commutative 94.6%

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

9. Applied egg-rr 94.6%

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

    1. unpow2 94.6%

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

    2. rem-sqrt-square 94.6%

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

    3. *-commutative 94.6%

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

11. Simplified 94.6%

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

12. Final simplification 94.6%

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

Alternative 3: 96.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(N[(N[Log[N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $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. Taylor expanded in f around 0 94.4%

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

   1. associate-/r* 94.4%

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

   2. distribute-rgt-out-- 94.4%

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

   3. metadata-eval 94.4%

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

4. Simplified 94.4%

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

5. Taylor expanded in f around 0 94.5%

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

   1. neg-mul-1 94.5%

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

   2. sub-neg 94.5%

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

   3. metadata-eval 94.5%

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

   4. associate-*l/ 94.5%

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

   5. log-div 94.5%

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

   6. associate-*l/ 94.5%

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

   7. metadata-eval 94.5%

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

7. Simplified 94.5%

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

8. Final simplification 94.5%

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

Alternative 4: 4.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \cdot \left(\pi \cdot \left(-{f}^{2}\right)\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* 0.08333333333333333 (* PI (- (pow f 2.0)))))

C

double code(double f) {
	return 0.08333333333333333 * (((double) M_PI) * -pow(f, 2.0));
}

Java

public static double code(double f) {
	return 0.08333333333333333 * (Math.PI * -Math.pow(f, 2.0));
}

Python

def code(f):
	return 0.08333333333333333 * (math.pi * -math.pow(f, 2.0))

Julia

function code(f)
	return Float64(0.08333333333333333 * Float64(pi * Float64(-(f ^ 2.0))))
end

MATLAB

function tmp = code(f)
	tmp = 0.08333333333333333 * (pi * -(f ^ 2.0));
end

Wolfram

code[f_] := N[(0.08333333333333333 * N[(Pi * (-N[Power[f, 2.0], $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. Taylor expanded in f around 0 94.9%

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

   1. +-commutative 94.9%

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

   2. associate-+l+ 94.9%

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

   3. fma-def 94.9%

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

   4. distribute-rgt-out-- 94.9%

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

   5. metadata-eval 94.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}\right) \]

   6. fma-def 94.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \color{blue}{\mathsf{fma}\left({f}^{5}, 8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}, f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}\right)}\right) \]

   7. distribute-rgt-out-- 94.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, \color{blue}{{\pi}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} - -8.138020833333333 \cdot 10^{-6}\right)}, f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right)}\right) \]

   8. metadata-eval 94.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}, f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right)}\right) \]

   9. distribute-rgt-out-- 94.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, f \cdot \color{blue}{\left(\pi \cdot \left(0.25 - -0.25\right)\right)}\right)\right)}\right) \]

4. Simplified 94.9%

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

5. Taylor expanded in f around 0 94.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right)\right)\right)} \]

7. Simplified 94.8%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\frac{\frac{4}{\pi}}{f}\right) + 0.5 \cdot \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right)\right)} \]

8. Taylor expanded in f around inf 4.2%

   \[\leadsto -\color{blue}{0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)} \]

9. Final simplification 4.2%

   \[\leadsto 0.08333333333333333 \cdot \left(\pi \cdot \left(-{f}^{2}\right)\right) \]

Reproduce

herbie shell --seed 2023313 
(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)))))))))