VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 96.5%

Time: 21.7s

Alternatives: 9

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, 1.0× speedup

Math

\[\begin{array}{l} \\ -\mathsf{fma}\left(2, \left({\pi}^{2} \cdot 0.041666666666666664\right) \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (fma
   2.0
   (* (* (pow PI 2.0) 0.041666666666666664) (/ (pow f 2.0) PI))
   (fma 4.0 (/ (- (log (/ 4.0 PI)) (log f)) PI) 0.0))))

C

double code(double f) {
	return -fma(2.0, ((pow(((double) M_PI), 2.0) * 0.041666666666666664) * (pow(f, 2.0) / ((double) M_PI))), fma(4.0, ((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI)), 0.0));
}

Julia

function code(f)
	return Float64(-fma(2.0, Float64(Float64((pi ^ 2.0) * 0.041666666666666664) * Float64((f ^ 2.0) / pi)), fma(4.0, Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi), 0.0)))
end

Wolfram

code[f_] := (-N[(2.0 * N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * N[(N[Power[f, 2.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] + 0.0), $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. Taylor expanded in f around 0 95.9%

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

4. Simplified 95.9%

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

   1. fma-udef 95.9%

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

   2. *-commutative 95.9%

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

   3. associate-/r/ 95.9%

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

   4. associate-*r/ 95.9%

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

   5. metadata-eval 95.9%

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

6. Applied egg-rr 95.9%

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

   1. fma-udef 95.9%

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

   2. associate-*r* 95.9%

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

   3. +-commutative 95.9%

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

   4. *-commutative 95.9%

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

   5. associate-/r/ 95.9%

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

   6. associate-*r* 95.9%

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

   7. metadata-eval 95.9%

      \[\leadsto -\mathsf{fma}\left(2, \left(\left(\pi \cdot \left(\left(0.005208333333333333 \cdot \color{blue}{-8}\right) \cdot \pi + 0.0625 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot 0.5 + 0\right) \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

   8. metadata-eval 95.9%

      \[\leadsto -\mathsf{fma}\left(2, \left(\left(\pi \cdot \left(\color{blue}{-0.041666666666666664} \cdot \pi + 0.0625 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot 0.5 + 0\right) \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

   9. *-commutative 95.9%

      \[\leadsto -\mathsf{fma}\left(2, \left(\left(\pi \cdot \left(-0.041666666666666664 \cdot \pi + 0.0625 \cdot \color{blue}{\left(2 \cdot \pi\right)}\right)\right) \cdot 0.5 + 0\right) \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

   10. associate-*r* 95.9%

       \[\leadsto -\mathsf{fma}\left(2, \left(\left(\pi \cdot \left(-0.041666666666666664 \cdot \pi + \color{blue}{\left(0.0625 \cdot 2\right) \cdot \pi}\right)\right) \cdot 0.5 + 0\right) \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

   11. metadata-eval 95.9%

       \[\leadsto -\mathsf{fma}\left(2, \left(\left(\pi \cdot \left(-0.041666666666666664 \cdot \pi + \color{blue}{0.125} \cdot \pi\right)\right) \cdot 0.5 + 0\right) \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

8. Applied egg-rr 95.9%

   \[\leadsto -\mathsf{fma}\left(2, \color{blue}{\left(\left(\pi \cdot \left(-0.041666666666666664 \cdot \pi + 0.125 \cdot \pi\right)\right) \cdot 0.5 + 0\right)} \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]
9. Step-by-step derivation

   1. +-rgt-identity 95.9%

      \[\leadsto -\mathsf{fma}\left(2, \color{blue}{\left(\left(\pi \cdot \left(-0.041666666666666664 \cdot \pi + 0.125 \cdot \pi\right)\right) \cdot 0.5\right)} \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

   2. associate-*r* 95.9%

      \[\leadsto -\mathsf{fma}\left(2, \color{blue}{\left(\pi \cdot \left(\left(-0.041666666666666664 \cdot \pi + 0.125 \cdot \pi\right) \cdot 0.5\right)\right)} \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

   3. *-commutative 95.9%

      \[\leadsto -\mathsf{fma}\left(2, \left(\pi \cdot \color{blue}{\left(0.5 \cdot \left(-0.041666666666666664 \cdot \pi + 0.125 \cdot \pi\right)\right)}\right) \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

   4. distribute-rgt-out 95.9%

      \[\leadsto -\mathsf{fma}\left(2, \left(\pi \cdot \left(0.5 \cdot \color{blue}{\left(\pi \cdot \left(-0.041666666666666664 + 0.125\right)\right)}\right)\right) \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

   5. metadata-eval 95.9%

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

10. Simplified 95.9%

    \[\leadsto -\mathsf{fma}\left(2, \color{blue}{\left(\pi \cdot \left(0.5 \cdot \left(\pi \cdot 0.08333333333333333\right)\right)\right)} \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]
11. Step-by-step derivation

    1. expm1-log1p-u 95.9%

       \[\leadsto -\mathsf{fma}\left(2, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.5 \cdot \left(\pi \cdot 0.08333333333333333\right)\right)\right)\right)} \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

    2. expm1-udef 95.9%

       \[\leadsto -\mathsf{fma}\left(2, \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(0.5 \cdot \left(\pi \cdot 0.08333333333333333\right)\right)\right)} - 1\right)} \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

    3. *-commutative 95.9%

       \[\leadsto -\mathsf{fma}\left(2, \left(e^{\mathsf{log1p}\left(\pi \cdot \left(0.5 \cdot \color{blue}{\left(0.08333333333333333 \cdot \pi\right)}\right)\right)} - 1\right) \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

    4. associate-*r* 95.9%

       \[\leadsto -\mathsf{fma}\left(2, \left(e^{\mathsf{log1p}\left(\pi \cdot \color{blue}{\left(\left(0.5 \cdot 0.08333333333333333\right) \cdot \pi\right)}\right)} - 1\right) \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

    5. metadata-eval 95.9%

       \[\leadsto -\mathsf{fma}\left(2, \left(e^{\mathsf{log1p}\left(\pi \cdot \left(\color{blue}{0.041666666666666664} \cdot \pi\right)\right)} - 1\right) \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

12. Applied egg-rr 95.9%

    \[\leadsto -\mathsf{fma}\left(2, \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(0.041666666666666664 \cdot \pi\right)\right)} - 1\right)} \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]
13. Step-by-step derivation

    1. expm1-def 95.9%

       \[\leadsto -\mathsf{fma}\left(2, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.041666666666666664 \cdot \pi\right)\right)\right)} \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

    2. expm1-log1p 95.9%

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

    3. *-commutative 95.9%

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

    4. associate-*r* 95.9%

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

    5. unpow2 95.9%

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

14. Simplified 95.9%

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

15. Final simplification 95.9%

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

Alternative 2: 96.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ -\mathsf{fma}\left(2, \frac{{f}^{2}}{\pi} \cdot \left(\pi \cdot \left(0.5 \cdot \left(\pi \cdot 0.08333333333333333\right)\right)\right), \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi}, 0\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (fma
   2.0
   (* (/ (pow f 2.0) PI) (* PI (* 0.5 (* PI 0.08333333333333333))))
   (fma 4.0 (/ (log (/ 4.0 (* PI f))) PI) 0.0))))

C

double code(double f) {
	return -fma(2.0, ((pow(f, 2.0) / ((double) M_PI)) * (((double) M_PI) * (0.5 * (((double) M_PI) * 0.08333333333333333)))), fma(4.0, (log((4.0 / (((double) M_PI) * f))) / ((double) M_PI)), 0.0));
}

Julia

function code(f)
	return Float64(-fma(2.0, Float64(Float64((f ^ 2.0) / pi) * Float64(pi * Float64(0.5 * Float64(pi * 0.08333333333333333)))), fma(4.0, Float64(log(Float64(4.0 / Float64(pi * f))) / pi), 0.0)))
end

Wolfram

code[f_] := (-N[(2.0 * N[(N[(N[Power[f, 2.0], $MachinePrecision] / Pi), $MachinePrecision] * N[(Pi * N[(0.5 * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] + 0.0), $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. Taylor expanded in f around 0 95.9%

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

4. Simplified 95.9%

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

   1. fma-udef 95.9%

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

   2. *-commutative 95.9%

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

   3. associate-/r/ 95.9%

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

   4. associate-*r/ 95.9%

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

   5. metadata-eval 95.9%

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

6. Applied egg-rr 95.9%

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

   1. fma-udef 95.9%

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

   2. associate-*r* 95.9%

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

   3. +-commutative 95.9%

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

   4. *-commutative 95.9%

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

   5. associate-/r/ 95.9%

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

   6. associate-*r* 95.9%

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

   7. metadata-eval 95.9%

      \[\leadsto -\mathsf{fma}\left(2, \left(\left(\pi \cdot \left(\left(0.005208333333333333 \cdot \color{blue}{-8}\right) \cdot \pi + 0.0625 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot 0.5 + 0\right) \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

   8. metadata-eval 95.9%

      \[\leadsto -\mathsf{fma}\left(2, \left(\left(\pi \cdot \left(\color{blue}{-0.041666666666666664} \cdot \pi + 0.0625 \cdot \left(\pi \cdot 2\right)\right)\right) \cdot 0.5 + 0\right) \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

   9. *-commutative 95.9%

      \[\leadsto -\mathsf{fma}\left(2, \left(\left(\pi \cdot \left(-0.041666666666666664 \cdot \pi + 0.0625 \cdot \color{blue}{\left(2 \cdot \pi\right)}\right)\right) \cdot 0.5 + 0\right) \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

   10. associate-*r* 95.9%

       \[\leadsto -\mathsf{fma}\left(2, \left(\left(\pi \cdot \left(-0.041666666666666664 \cdot \pi + \color{blue}{\left(0.0625 \cdot 2\right) \cdot \pi}\right)\right) \cdot 0.5 + 0\right) \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

   11. metadata-eval 95.9%

       \[\leadsto -\mathsf{fma}\left(2, \left(\left(\pi \cdot \left(-0.041666666666666664 \cdot \pi + \color{blue}{0.125} \cdot \pi\right)\right) \cdot 0.5 + 0\right) \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

8. Applied egg-rr 95.9%

   \[\leadsto -\mathsf{fma}\left(2, \color{blue}{\left(\left(\pi \cdot \left(-0.041666666666666664 \cdot \pi + 0.125 \cdot \pi\right)\right) \cdot 0.5 + 0\right)} \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]
9. Step-by-step derivation

   1. +-rgt-identity 95.9%

      \[\leadsto -\mathsf{fma}\left(2, \color{blue}{\left(\left(\pi \cdot \left(-0.041666666666666664 \cdot \pi + 0.125 \cdot \pi\right)\right) \cdot 0.5\right)} \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

   2. associate-*r* 95.9%

      \[\leadsto -\mathsf{fma}\left(2, \color{blue}{\left(\pi \cdot \left(\left(-0.041666666666666664 \cdot \pi + 0.125 \cdot \pi\right) \cdot 0.5\right)\right)} \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

   3. *-commutative 95.9%

      \[\leadsto -\mathsf{fma}\left(2, \left(\pi \cdot \color{blue}{\left(0.5 \cdot \left(-0.041666666666666664 \cdot \pi + 0.125 \cdot \pi\right)\right)}\right) \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

   4. distribute-rgt-out 95.9%

      \[\leadsto -\mathsf{fma}\left(2, \left(\pi \cdot \left(0.5 \cdot \color{blue}{\left(\pi \cdot \left(-0.041666666666666664 + 0.125\right)\right)}\right)\right) \cdot \frac{{f}^{2}}{\pi}, \mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, 0\right)\right) \]

   5. metadata-eval 95.9%

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

10. Simplified 95.9%

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

11. Taylor expanded in f around inf 95.9%

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

    1. cancel-sign-sub-inv 95.9%

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

    2. *-lft-identity 95.9%

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

    3. metadata-eval 95.9%

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

    4. distribute-lft-in 95.9%

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

    5. +-commutative 95.9%

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

    6. *-lft-identity 95.9%

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

    7. +-commutative 95.9%

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

    8. log-prod 95.9%

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

    9. associate-*r/ 95.9%

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

    10. *-rgt-identity 95.9%

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

    11. associate-/r* 95.9%

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

13. Simplified 95.9%

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

14. Final simplification 95.9%

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

Alternative 3: 96.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(0.0625, 2 \cdot \pi, \frac{-2}{\frac{0.5 \cdot \frac{0.5}{\pi}}{0.005208333333333333}}\right), \frac{\frac{4}{\pi}}{f}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (fma
    f
    (fma
     0.0625
     (* 2.0 PI)
     (/ -2.0 (/ (* 0.5 (/ 0.5 PI)) 0.005208333333333333)))
    (/ (/ 4.0 PI) f)))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(fma(f, fma(0.0625, (2.0 * ((double) M_PI)), (-2.0 / ((0.5 * (0.5 / ((double) M_PI))) / 0.005208333333333333))), ((4.0 / ((double) M_PI)) / f))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(fma(f, fma(0.0625, Float64(2.0 * pi), Float64(-2.0 / Float64(Float64(0.5 * Float64(0.5 / pi)) / 0.005208333333333333))), Float64(Float64(4.0 / pi) / f))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(0.0625 * N[(2.0 * Pi), $MachinePrecision] + N[(-2.0 / N[(N[(0.5 * N[(0.5 / Pi), $MachinePrecision]), $MachinePrecision] / 0.005208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $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. Taylor expanded in f around 0 95.7%

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

4. Simplified 95.7%

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

5. Final simplification 95.7%

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

Alternative 4: 96.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(N[(4.0 * N[(N[(N[Log[f], $MachinePrecision] - N[Log[N[(4.0 / Pi), $MachinePrecision]], $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. Taylor expanded in f around 0 95.2%

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

   1. distribute-rgt-out-- 95.2%

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

   2. metadata-eval 95.2%

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

4. Simplified 95.2%

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

5. Taylor expanded in f around 0 95.4%

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

6. Final simplification 95.4%

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

Alternative 5: 96.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{-\log \left(4 \cdot \left(0.03125 \cdot \left(\pi \cdot f\right) + \frac{1}{\pi \cdot f}\right)\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* 4.0 (/ (- (log (* 4.0 (+ (* 0.03125 (* PI f)) (/ 1.0 (* PI f)))))) PI)))

C

double code(double f) {
	return 4.0 * (-log((4.0 * ((0.03125 * (((double) M_PI) * f)) + (1.0 / (((double) M_PI) * f))))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return 4.0 * (-Math.log((4.0 * ((0.03125 * (Math.PI * f)) + (1.0 / (Math.PI * f))))) / Math.PI);
}

Python

def code(f):
	return 4.0 * (-math.log((4.0 * ((0.03125 * (math.pi * f)) + (1.0 / (math.pi * f))))) / math.pi)

Julia

function code(f)
	return Float64(4.0 * Float64(Float64(-log(Float64(4.0 * Float64(Float64(0.03125 * Float64(pi * f)) + Float64(1.0 / Float64(pi * f)))))) / pi))
end

MATLAB

function tmp = code(f)
	tmp = 4.0 * (-log((4.0 * ((0.03125 * (pi * f)) + (1.0 / (pi * f))))) / pi);
end

Wolfram

code[f_] := N[(4.0 * N[((-N[Log[N[(4.0 * N[(N[(0.03125 * N[(Pi * f), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. Taylor expanded in f around 0 95.2%

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

   1. distribute-rgt-out-- 95.2%

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

   2. metadata-eval 95.2%

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

4. Simplified 95.2%

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

   1. associate-*l/ 95.3%

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

   2. *-un-lft-identity 95.3%

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

   3. cosh-undef 95.3%

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

   4. *-commutative 95.3%

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

   5. div-inv 95.3%

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

   6. metadata-eval 95.3%

      \[\leadsto -\frac{\log \left(\frac{2 \cdot \cosh \left(f \cdot \left(\pi \cdot \color{blue}{0.25}\right)\right)}{f \cdot \left(\pi \cdot 0.5\right)}\right)}{\frac{\pi}{4}} \]

   7. div-inv 95.3%

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

   8. metadata-eval 95.3%

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

6. Applied egg-rr 95.3%

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

   1. *-lft-identity 95.3%

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

   2. *-commutative 95.3%

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

   3. times-frac 95.3%

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

   4. metadata-eval 95.3%

      \[\leadsto -\color{blue}{4} \cdot \frac{\log \left(\frac{2 \cdot \cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{f \cdot \left(\pi \cdot 0.5\right)}\right)}{\pi} \]

   5. associate-*r* 95.3%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{2 \cdot \cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\color{blue}{\left(f \cdot \pi\right) \cdot 0.5}}\right)}{\pi} \]

   6. *-commutative 95.3%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{2 \cdot \cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\color{blue}{0.5 \cdot \left(f \cdot \pi\right)}}\right)}{\pi} \]

   7. times-frac 95.3%

      \[\leadsto -4 \cdot \frac{\log \color{blue}{\left(\frac{2}{0.5} \cdot \frac{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{f \cdot \pi}\right)}}{\pi} \]

   8. metadata-eval 95.3%

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

   9. *-commutative 95.3%

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

   10. associate-*l* 95.3%

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

8. Simplified 95.3%

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

9. Taylor expanded in f around 0 95.4%

   \[\leadsto -4 \cdot \frac{\log \left(4 \cdot \color{blue}{\left(0.03125 \cdot \left(f \cdot \pi\right) + \frac{1}{f \cdot \pi}\right)}\right)}{\pi} \]

10. Final simplification 95.4%

    \[\leadsto 4 \cdot \frac{-\log \left(4 \cdot \left(0.03125 \cdot \left(\pi \cdot f\right) + \frac{1}{\pi \cdot f}\right)\right)}{\pi} \]

Alternative 6: 96.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(4.0 * N[(N[(N[Log[f], $MachinePrecision] - N[Log[N[(2.0 / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $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. Taylor expanded in f around 0 95.2%

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

   1. distribute-rgt-out-- 95.2%

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

   2. metadata-eval 95.2%

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

   3. mul-1-neg 95.2%

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

4. Simplified 95.2%

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

5. Final simplification 95.2%

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

Alternative 7: 95.8% accurate, 3.3× speedup

Math

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

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) * Float64(-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 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. Taylor expanded in f around 0 95.2%

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

   1. associate-*r/ 95.2%

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

   2. associate-/l* 95.0%

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

   3. mul-1-neg 95.0%

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

   4. unsub-neg 95.0%

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

   5. distribute-rgt-out-- 95.0%

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

   6. metadata-eval 95.0%

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

4. Simplified 95.0%

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

   1. add-sqr-sqrt 94.5%

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

   2. sqrt-unprod 95.1%

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

   3. pow2 95.1%

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

   4. diff-log 95.1%

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

6. Applied egg-rr 95.1%

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

8. Simplified 95.1%

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

9. Taylor expanded in f around 0 95.1%

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

    1. fabs-div 95.1%

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

    2. *-commutative 95.1%

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

    3. fabs-div 95.1%

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

    4. rem-square-sqrt 94.5%

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

    5. fabs-sqr 94.5%

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

    6. rem-square-sqrt 95.0%

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

    7. associate-/r/ 95.0%

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

    8. associate-/r* 95.0%

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

11. Simplified 95.0%

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

12. Final simplification 95.0%

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

Alternative 8: 95.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(\frac{4}{\pi \cdot f}\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (- (log (/ 4.0 (* PI f)))) (* PI 0.25)))

C

double code(double f) {
	return -log((4.0 / (((double) M_PI) * f))) / (((double) M_PI) * 0.25);
}

Java

public static double code(double f) {
	return -Math.log((4.0 / (Math.PI * f))) / (Math.PI * 0.25);
}

Python

def code(f):
	return -math.log((4.0 / (math.pi * f))) / (math.pi * 0.25)

Julia

function code(f)
	return Float64(Float64(-log(Float64(4.0 / Float64(pi * f)))) / Float64(pi * 0.25))
end

MATLAB

function tmp = code(f)
	tmp = -log((4.0 / (pi * f))) / (pi * 0.25);
end

Wolfram

code[f_] := N[((-N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $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. Taylor expanded in f around 0 95.0%

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

   1. associate-/l/ 95.0%

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

   2. distribute-rgt-out-- 95.0%

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

   3. *-commutative 95.0%

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

   4. associate-/r* 95.0%

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

   5. metadata-eval 95.0%

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

   6. metadata-eval 95.0%

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

4. Simplified 95.0%

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

   1. associate-*l/ 95.2%

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

   2. *-un-lft-identity 95.2%

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

   3. associate-/l/ 95.2%

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

   4. div-inv 95.2%

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

   5. metadata-eval 95.2%

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

6. Applied egg-rr 95.2%

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

7. Final simplification 95.2%

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

Alternative 9: 1.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

3. Applied egg-rr 1.7%

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

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.5}{\pi}} \]
5. Step-by-step derivation

   1. associate-*r/ 1.6%

      \[\leadsto -\color{blue}{\frac{4 \cdot \log 0.5}{\pi}} \]

6. Simplified 1.6%

   \[\leadsto -\color{blue}{\frac{4 \cdot \log 0.5}{\pi}} \]

7. Final simplification 1.6%

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

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)))))))))