VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 96.3%

Time: 43.2s

Alternatives: 7

Speedup: 3.3×

• could not determine a ground truth

  1. f = -3.597398395093695e+262

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: 96.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{f \cdot \left(f \cdot \left(\pi \cdot \left(0.5 \cdot \left(\left(\pi \cdot 0.5\right) \cdot 0.08333333333333333\right)\right)\right)\right) + \log \left(\frac{\frac{4}{f}}{\pi}\right)}{\pi} \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/
   (+
    (* f (* f (* PI (* 0.5 (* (* PI 0.5) 0.08333333333333333)))))
    (log (/ (/ 4.0 f) PI)))
   PI)
  (- 4.0)))

C

double code(double f) {
	return (((f * (f * (((double) M_PI) * (0.5 * ((((double) M_PI) * 0.5) * 0.08333333333333333))))) + log(((4.0 / f) / ((double) M_PI)))) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return (((f * (f * (Math.PI * (0.5 * ((Math.PI * 0.5) * 0.08333333333333333))))) + Math.log(((4.0 / f) / Math.PI))) / Math.PI) * -4.0;
}

Python

def code(f):
	return (((f * (f * (math.pi * (0.5 * ((math.pi * 0.5) * 0.08333333333333333))))) + math.log(((4.0 / f) / math.pi))) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(Float64(Float64(f * Float64(f * Float64(pi * Float64(0.5 * Float64(Float64(pi * 0.5) * 0.08333333333333333))))) + log(Float64(Float64(4.0 / f) / pi))) / pi) * Float64(-4.0))
end

MATLAB

function tmp = code(f)
	tmp = (((f * (f * (pi * (0.5 * ((pi * 0.5) * 0.08333333333333333))))) + log(((4.0 / f) / pi))) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[(N[(f * N[(f * N[(Pi * N[(0.5 * N[(N[(Pi * 0.5), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Log[N[(N[(4.0 / f), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision]

Derivation

1. Initial program 7.2%

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

2. Taylor expanded in f around 0 96.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \left(-1 \cdot \log f + \left(0.5 \cdot \left(f \cdot \left(\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot \left({\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2} \cdot {\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}\right) + \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right) \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right)\right)\right)\right)} \]

4. Simplified 96.5%

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

   1. *-un-lft-identity 96.5%

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

   2. fma-udef 96.5%

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

   3. div-inv 96.5%

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

   4. metadata-eval 96.5%

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

   5. associate-*l/ 96.5%

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

   6. div-inv 96.5%

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

   7. metadata-eval 96.5%

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

6. Applied egg-rr 96.5%

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

   1. expm1-log1p-u 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.0625 \cdot \left(\pi \cdot 2\right) + \frac{{\pi}^{3} \cdot -2}{\left(0.25 \cdot {\pi}^{2}\right) \cdot 192}\right)\right)}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   2. expm1-udef 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{e^{\mathsf{log1p}\left(0.0625 \cdot \left(\pi \cdot 2\right) + \frac{{\pi}^{3} \cdot -2}{\left(0.25 \cdot {\pi}^{2}\right) \cdot 192}\right)} - 1}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   3. fma-def 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.0625, \pi \cdot 2, \frac{{\pi}^{3} \cdot -2}{\left(0.25 \cdot {\pi}^{2}\right) \cdot 192}\right)}\right)} - 1, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   4. times-frac 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.0625, \pi \cdot 2, \color{blue}{\frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot \frac{-2}{192}}\right)\right)} - 1, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   5. metadata-eval 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.0625, \pi \cdot 2, \frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot \color{blue}{-0.010416666666666666}\right)\right)} - 1, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

8. Applied egg-rr 96.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.0625, \pi \cdot 2, \frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot -0.010416666666666666\right)\right)} - 1}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]
9. Step-by-step derivation

   1. expm1-def 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.0625, \pi \cdot 2, \frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot -0.010416666666666666\right)\right)\right)}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   2. expm1-log1p 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\mathsf{fma}\left(0.0625, \pi \cdot 2, \frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot -0.010416666666666666\right)}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   3. fma-udef 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{0.0625 \cdot \left(\pi \cdot 2\right) + \frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot -0.010416666666666666}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   4. +-commutative 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot -0.010416666666666666 + 0.0625 \cdot \left(\pi \cdot 2\right)}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   5. fma-udef 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\mathsf{fma}\left(\frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}}, -0.010416666666666666, 0.0625 \cdot \left(\pi \cdot 2\right)\right)}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   6. unpow3 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(\frac{\color{blue}{\left(\pi \cdot \pi\right) \cdot \pi}}{0.25 \cdot {\pi}^{2}}, -0.010416666666666666, 0.0625 \cdot \left(\pi \cdot 2\right)\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   7. unpow2 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(\frac{\color{blue}{{\pi}^{2}} \cdot \pi}{0.25 \cdot {\pi}^{2}}, -0.010416666666666666, 0.0625 \cdot \left(\pi \cdot 2\right)\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   8. *-commutative 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(\frac{{\pi}^{2} \cdot \pi}{\color{blue}{{\pi}^{2} \cdot 0.25}}, -0.010416666666666666, 0.0625 \cdot \left(\pi \cdot 2\right)\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   9. times-frac 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(\color{blue}{\frac{{\pi}^{2}}{{\pi}^{2}} \cdot \frac{\pi}{0.25}}, -0.010416666666666666, 0.0625 \cdot \left(\pi \cdot 2\right)\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   10. *-inverses 96.5%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(\color{blue}{1} \cdot \frac{\pi}{0.25}, -0.010416666666666666, 0.0625 \cdot \left(\pi \cdot 2\right)\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   11. *-commutative 96.5%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(1 \cdot \frac{\pi}{0.25}, -0.010416666666666666, \color{blue}{\left(\pi \cdot 2\right) \cdot 0.0625}\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   12. associate-*l* 96.5%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(1 \cdot \frac{\pi}{0.25}, -0.010416666666666666, \color{blue}{\pi \cdot \left(2 \cdot 0.0625\right)}\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   13. metadata-eval 96.5%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(1 \cdot \frac{\pi}{0.25}, -0.010416666666666666, \pi \cdot \color{blue}{0.125}\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

10. Simplified 96.5%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\mathsf{fma}\left(1 \cdot \frac{\pi}{0.25}, -0.010416666666666666, \pi \cdot 0.125\right)}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]
11. Step-by-step derivation

    1. clear-num 96.5%

       \[\leadsto -\color{blue}{\frac{4}{\pi}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(1 \cdot \frac{\pi}{0.25}, -0.010416666666666666, \pi \cdot 0.125\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

    2. distribute-lft-in 96.3%

       \[\leadsto -\color{blue}{\left(\frac{4}{\pi} \cdot \log \left(\frac{2}{\pi \cdot 0.5}\right) + \frac{4}{\pi} \cdot \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(1 \cdot \frac{\pi}{0.25}, -0.010416666666666666, \pi \cdot 0.125\right), 0\right) \cdot 0.5\right) - \log f\right)\right)} \]

    3. associate-/r* 96.3%

       \[\leadsto -\left(\frac{4}{\pi} \cdot \log \color{blue}{\left(\frac{\frac{2}{\pi}}{0.5}\right)} + \frac{4}{\pi} \cdot \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(1 \cdot \frac{\pi}{0.25}, -0.010416666666666666, \pi \cdot 0.125\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

12. Applied egg-rr 96.3%

    \[\leadsto -\color{blue}{\left(\frac{4}{\pi} \cdot \log \left(\frac{\frac{2}{\pi}}{0.5}\right) + \frac{4}{\pi} \cdot \left(\left(f \cdot f\right) \cdot \left(0.5 \cdot \mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(\pi \cdot 4, -0.010416666666666666, \pi \cdot 0.125\right), 0\right)\right) - \log f\right)\right)} \]

14. Simplified 96.5%

    \[\leadsto -\color{blue}{4 \cdot \frac{f \cdot \left(f \cdot \left(\pi \cdot \left(\left(\left(\pi \cdot 0.5\right) \cdot 0.08333333333333333\right) \cdot 0.5\right)\right)\right) + \log \left(\frac{\frac{4}{f}}{\pi}\right)}{\pi}} \]

15. Final simplification 96.5%

    \[\leadsto \frac{f \cdot \left(f \cdot \left(\pi \cdot \left(0.5 \cdot \left(\left(\pi \cdot 0.5\right) \cdot 0.08333333333333333\right)\right)\right)\right) + \log \left(\frac{\frac{4}{f}}{\pi}\right)}{\pi} \cdot \left(-4\right) \]

Alternative 2: 96.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

2. Taylor expanded in f around 0 96.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \left(-1 \cdot \log f + \left(0.5 \cdot \left(f \cdot \left(\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot \left({\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2} \cdot {\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}\right) + \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right) \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right)\right)\right)\right)} \]

4. Simplified 96.5%

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

   1. *-un-lft-identity 96.5%

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

   2. fma-udef 96.5%

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

   3. div-inv 96.5%

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

   4. metadata-eval 96.5%

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

   5. associate-*l/ 96.5%

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

   6. div-inv 96.5%

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

   7. metadata-eval 96.5%

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

6. Applied egg-rr 96.5%

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

   1. expm1-log1p-u 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.0625 \cdot \left(\pi \cdot 2\right) + \frac{{\pi}^{3} \cdot -2}{\left(0.25 \cdot {\pi}^{2}\right) \cdot 192}\right)\right)}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   2. expm1-udef 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{e^{\mathsf{log1p}\left(0.0625 \cdot \left(\pi \cdot 2\right) + \frac{{\pi}^{3} \cdot -2}{\left(0.25 \cdot {\pi}^{2}\right) \cdot 192}\right)} - 1}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   3. fma-def 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.0625, \pi \cdot 2, \frac{{\pi}^{3} \cdot -2}{\left(0.25 \cdot {\pi}^{2}\right) \cdot 192}\right)}\right)} - 1, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   4. times-frac 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.0625, \pi \cdot 2, \color{blue}{\frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot \frac{-2}{192}}\right)\right)} - 1, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   5. metadata-eval 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.0625, \pi \cdot 2, \frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot \color{blue}{-0.010416666666666666}\right)\right)} - 1, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

8. Applied egg-rr 96.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.0625, \pi \cdot 2, \frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot -0.010416666666666666\right)\right)} - 1}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]
9. Step-by-step derivation

   1. expm1-def 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.0625, \pi \cdot 2, \frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot -0.010416666666666666\right)\right)\right)}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   2. expm1-log1p 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\mathsf{fma}\left(0.0625, \pi \cdot 2, \frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot -0.010416666666666666\right)}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   3. fma-udef 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{0.0625 \cdot \left(\pi \cdot 2\right) + \frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot -0.010416666666666666}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   4. +-commutative 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot -0.010416666666666666 + 0.0625 \cdot \left(\pi \cdot 2\right)}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   5. fma-udef 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\mathsf{fma}\left(\frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}}, -0.010416666666666666, 0.0625 \cdot \left(\pi \cdot 2\right)\right)}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   6. unpow3 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(\frac{\color{blue}{\left(\pi \cdot \pi\right) \cdot \pi}}{0.25 \cdot {\pi}^{2}}, -0.010416666666666666, 0.0625 \cdot \left(\pi \cdot 2\right)\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   7. unpow2 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(\frac{\color{blue}{{\pi}^{2}} \cdot \pi}{0.25 \cdot {\pi}^{2}}, -0.010416666666666666, 0.0625 \cdot \left(\pi \cdot 2\right)\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   8. *-commutative 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(\frac{{\pi}^{2} \cdot \pi}{\color{blue}{{\pi}^{2} \cdot 0.25}}, -0.010416666666666666, 0.0625 \cdot \left(\pi \cdot 2\right)\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   9. times-frac 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(\color{blue}{\frac{{\pi}^{2}}{{\pi}^{2}} \cdot \frac{\pi}{0.25}}, -0.010416666666666666, 0.0625 \cdot \left(\pi \cdot 2\right)\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   10. *-inverses 96.5%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(\color{blue}{1} \cdot \frac{\pi}{0.25}, -0.010416666666666666, 0.0625 \cdot \left(\pi \cdot 2\right)\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   11. *-commutative 96.5%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(1 \cdot \frac{\pi}{0.25}, -0.010416666666666666, \color{blue}{\left(\pi \cdot 2\right) \cdot 0.0625}\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   12. associate-*l* 96.5%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(1 \cdot \frac{\pi}{0.25}, -0.010416666666666666, \color{blue}{\pi \cdot \left(2 \cdot 0.0625\right)}\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   13. metadata-eval 96.5%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(1 \cdot \frac{\pi}{0.25}, -0.010416666666666666, \pi \cdot \color{blue}{0.125}\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

10. Simplified 96.5%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\mathsf{fma}\left(1 \cdot \frac{\pi}{0.25}, -0.010416666666666666, \pi \cdot 0.125\right)}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

11. Taylor expanded in f around 0 96.5%

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

    1. fma-def 96.5%

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

    2. log-div 96.5%

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

    3. associate-/r* 96.5%

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

    4. associate-/l/ 96.5%

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

    5. unpow2 96.5%

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

    6. *-commutative 96.5%

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

    7. distribute-rgt-out 96.5%

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

    8. associate-*l* 96.5%

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

    9. metadata-eval 96.5%

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

13. Simplified 96.5%

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

14. Final simplification 96.5%

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

Alternative 3: 96.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. *-commutative 7.2%

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

   2. distribute-rgt-neg-in 7.2%

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

3. Simplified 7.2%

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

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. *-commutative 95.9%

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

   2. associate-*l/ 95.9%

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

   3. mul-1-neg 95.9%

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

   4. unsub-neg 95.9%

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

   5. distribute-rgt-out-- 95.9%

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

   6. metadata-eval 95.9%

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

6. Simplified 95.9%

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

7. Taylor expanded in f around 0 95.9%

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

   1. div-sub 95.8%

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

   2. metadata-eval 95.8%

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

   3. associate-/r* 95.8%

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

   4. *-commutative 95.8%

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

   5. div-sub 95.9%

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

   6. log-div 95.9%

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

   7. *-commutative 95.9%

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

   8. associate-/r* 95.9%

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

   9. metadata-eval 95.9%

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

   10. associate-/r* 95.9%

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

9. Simplified 95.9%

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

10. Taylor expanded in f around inf 95.9%

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

11. Final simplification 95.9%

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

Alternative 4: 96.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. *-commutative 7.2%

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

   2. distribute-rgt-neg-in 7.2%

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

3. Simplified 7.2%

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

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. *-commutative 95.9%

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

   2. associate-*l/ 95.9%

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

   3. mul-1-neg 95.9%

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

   4. unsub-neg 95.9%

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

   5. distribute-rgt-out-- 95.9%

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

   6. metadata-eval 95.9%

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

6. Simplified 95.9%

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

   1. log-div 95.9%

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

8. Applied egg-rr 95.9%

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

   1. log-div 95.9%

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

   2. *-commutative 95.9%

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

   3. associate-/r* 95.9%

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

   4. metadata-eval 95.9%

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

10. Simplified 95.9%

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

11. Final simplification 95.9%

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

Alternative 5: 96.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. *-commutative 7.2%

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

   2. distribute-rgt-neg-in 7.2%

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

3. Simplified 7.2%

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

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. *-commutative 95.9%

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

   2. associate-*l/ 95.9%

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

   3. mul-1-neg 95.9%

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

   4. unsub-neg 95.9%

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

   5. distribute-rgt-out-- 95.9%

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

   6. metadata-eval 95.9%

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

6. Simplified 95.9%

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

7. Taylor expanded in f around 0 95.9%

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

   1. div-sub 95.8%

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

   2. metadata-eval 95.8%

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

   3. associate-/r* 95.8%

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

   4. *-commutative 95.8%

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

   5. div-sub 95.9%

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

   6. log-div 95.9%

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

   7. *-commutative 95.9%

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

   8. associate-/r* 95.9%

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

   9. metadata-eval 95.9%

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

   10. associate-/r* 95.9%

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

9. Simplified 95.9%

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

10. Final simplification 95.9%

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

Alternative 6: 96.0% accurate, 3.3× speedup

Math

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

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(-4.0 * Float64(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[(-4.0 * N[(N[Log[N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.2%

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

   1. *-commutative 7.2%

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

   2. distribute-rgt-neg-in 7.2%

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

3. Simplified 7.2%

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

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. *-commutative 95.9%

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

   2. associate-*l/ 95.9%

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

   3. mul-1-neg 95.9%

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

   4. unsub-neg 95.9%

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

   5. distribute-rgt-out-- 95.9%

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

   6. metadata-eval 95.9%

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

6. Simplified 95.9%

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

7. Taylor expanded in f around 0 95.9%

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

   1. div-sub 95.8%

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

   2. metadata-eval 95.8%

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

   3. associate-/r* 95.8%

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

   4. *-commutative 95.8%

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

   5. div-sub 95.9%

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

   6. log-div 95.9%

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

   7. *-commutative 95.9%

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

   8. associate-/r* 95.9%

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

   9. metadata-eval 95.9%

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

   10. associate-/r* 95.9%

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

9. Simplified 95.9%

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

10. Taylor expanded in f around 0 95.9%

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

    1. mul-1-neg 95.9%

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

    2. unsub-neg 95.9%

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

    3. log-div 95.9%

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

    4. associate--r+ 95.7%

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

    5. log-prod 95.8%

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

    6. log-div 95.9%

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

    7. rem-log-exp 95.9%

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

    8. *-rgt-identity 95.9%

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

    9. associate-*r/ 95.7%

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

    10. exp-to-pow 95.7%

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

    11. metadata-eval 95.7%

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

    12. associate-/r* 95.7%

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

    13. *-commutative 95.7%

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

    14. associate-*r* 95.7%

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

12. Simplified 95.9%

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

13. Final simplification 95.9%

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

Alternative 7: 4.2% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

2. Taylor expanded in f around 0 96.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \left(-1 \cdot \log f + \left(0.5 \cdot \left(f \cdot \left(\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot \left({\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2} \cdot {\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}\right) + \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right) \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)\right)\right)\right)\right)} \]

4. Simplified 96.5%

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

   1. *-un-lft-identity 96.5%

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

   2. fma-udef 96.5%

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

   3. div-inv 96.5%

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

   4. metadata-eval 96.5%

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

   5. associate-*l/ 96.5%

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

   6. div-inv 96.5%

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

   7. metadata-eval 96.5%

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

6. Applied egg-rr 96.5%

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

   1. expm1-log1p-u 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.0625 \cdot \left(\pi \cdot 2\right) + \frac{{\pi}^{3} \cdot -2}{\left(0.25 \cdot {\pi}^{2}\right) \cdot 192}\right)\right)}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   2. expm1-udef 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{e^{\mathsf{log1p}\left(0.0625 \cdot \left(\pi \cdot 2\right) + \frac{{\pi}^{3} \cdot -2}{\left(0.25 \cdot {\pi}^{2}\right) \cdot 192}\right)} - 1}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   3. fma-def 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.0625, \pi \cdot 2, \frac{{\pi}^{3} \cdot -2}{\left(0.25 \cdot {\pi}^{2}\right) \cdot 192}\right)}\right)} - 1, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   4. times-frac 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.0625, \pi \cdot 2, \color{blue}{\frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot \frac{-2}{192}}\right)\right)} - 1, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   5. metadata-eval 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.0625, \pi \cdot 2, \frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot \color{blue}{-0.010416666666666666}\right)\right)} - 1, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

8. Applied egg-rr 96.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.0625, \pi \cdot 2, \frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot -0.010416666666666666\right)\right)} - 1}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]
9. Step-by-step derivation

   1. expm1-def 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.0625, \pi \cdot 2, \frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot -0.010416666666666666\right)\right)\right)}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   2. expm1-log1p 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\mathsf{fma}\left(0.0625, \pi \cdot 2, \frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot -0.010416666666666666\right)}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   3. fma-udef 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{0.0625 \cdot \left(\pi \cdot 2\right) + \frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot -0.010416666666666666}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   4. +-commutative 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}} \cdot -0.010416666666666666 + 0.0625 \cdot \left(\pi \cdot 2\right)}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   5. fma-udef 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\mathsf{fma}\left(\frac{{\pi}^{3}}{0.25 \cdot {\pi}^{2}}, -0.010416666666666666, 0.0625 \cdot \left(\pi \cdot 2\right)\right)}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   6. unpow3 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(\frac{\color{blue}{\left(\pi \cdot \pi\right) \cdot \pi}}{0.25 \cdot {\pi}^{2}}, -0.010416666666666666, 0.0625 \cdot \left(\pi \cdot 2\right)\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   7. unpow2 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(\frac{\color{blue}{{\pi}^{2}} \cdot \pi}{0.25 \cdot {\pi}^{2}}, -0.010416666666666666, 0.0625 \cdot \left(\pi \cdot 2\right)\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   8. *-commutative 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(\frac{{\pi}^{2} \cdot \pi}{\color{blue}{{\pi}^{2} \cdot 0.25}}, -0.010416666666666666, 0.0625 \cdot \left(\pi \cdot 2\right)\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   9. times-frac 96.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(\color{blue}{\frac{{\pi}^{2}}{{\pi}^{2}} \cdot \frac{\pi}{0.25}}, -0.010416666666666666, 0.0625 \cdot \left(\pi \cdot 2\right)\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   10. *-inverses 96.5%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(\color{blue}{1} \cdot \frac{\pi}{0.25}, -0.010416666666666666, 0.0625 \cdot \left(\pi \cdot 2\right)\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   11. *-commutative 96.5%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(1 \cdot \frac{\pi}{0.25}, -0.010416666666666666, \color{blue}{\left(\pi \cdot 2\right) \cdot 0.0625}\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   12. associate-*l* 96.5%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(1 \cdot \frac{\pi}{0.25}, -0.010416666666666666, \color{blue}{\pi \cdot \left(2 \cdot 0.0625\right)}\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

   13. metadata-eval 96.5%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(1 \cdot \frac{\pi}{0.25}, -0.010416666666666666, \pi \cdot \color{blue}{0.125}\right), 0\right) \cdot 0.5\right) - \log f\right)\right) \]

10. Simplified 96.5%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) + \left(\left(f \cdot f\right) \cdot \left(\mathsf{fma}\left(\pi \cdot 0.5, \color{blue}{\mathsf{fma}\left(1 \cdot \frac{\pi}{0.25}, -0.010416666666666666, \pi \cdot 0.125\right)}, 0\right) \cdot 0.5\right) - \log f\right)\right) \]

11. Taylor expanded in f around inf 4.3%

    \[\leadsto -\color{blue}{{f}^{2} \cdot \left(-0.041666666666666664 \cdot \pi + 0.125 \cdot \pi\right)} \]
12. Step-by-step derivation

    1. unpow2 4.3%

       \[\leadsto -\color{blue}{\left(f \cdot f\right)} \cdot \left(-0.041666666666666664 \cdot \pi + 0.125 \cdot \pi\right) \]

    2. *-commutative 4.3%

       \[\leadsto -\left(f \cdot f\right) \cdot \left(\color{blue}{\pi \cdot -0.041666666666666664} + 0.125 \cdot \pi\right) \]

    3. metadata-eval 4.3%

       \[\leadsto -\left(f \cdot f\right) \cdot \left(\pi \cdot \color{blue}{\left(4 \cdot -0.010416666666666666\right)} + 0.125 \cdot \pi\right) \]

    4. associate-*l* 4.3%

       \[\leadsto -\left(f \cdot f\right) \cdot \left(\color{blue}{\left(\pi \cdot 4\right) \cdot -0.010416666666666666} + 0.125 \cdot \pi\right) \]

    5. fma-def 4.3%

       \[\leadsto -\left(f \cdot f\right) \cdot \color{blue}{\mathsf{fma}\left(\pi \cdot 4, -0.010416666666666666, 0.125 \cdot \pi\right)} \]

    6. *-commutative 4.3%

       \[\leadsto -\left(f \cdot f\right) \cdot \mathsf{fma}\left(\pi \cdot 4, -0.010416666666666666, \color{blue}{\pi \cdot 0.125}\right) \]

    7. associate-*l* 4.3%

       \[\leadsto -\color{blue}{f \cdot \left(f \cdot \mathsf{fma}\left(\pi \cdot 4, -0.010416666666666666, \pi \cdot 0.125\right)\right)} \]

    8. fma-def 4.3%

       \[\leadsto -f \cdot \left(f \cdot \color{blue}{\left(\left(\pi \cdot 4\right) \cdot -0.010416666666666666 + \pi \cdot 0.125\right)}\right) \]

    9. associate-*l* 4.3%

       \[\leadsto -f \cdot \left(f \cdot \left(\color{blue}{\pi \cdot \left(4 \cdot -0.010416666666666666\right)} + \pi \cdot 0.125\right)\right) \]

    10. metadata-eval 4.3%

        \[\leadsto -f \cdot \left(f \cdot \left(\pi \cdot \color{blue}{-0.041666666666666664} + \pi \cdot 0.125\right)\right) \]

    11. distribute-lft-out 4.3%

        \[\leadsto -f \cdot \left(f \cdot \color{blue}{\left(\pi \cdot \left(-0.041666666666666664 + 0.125\right)\right)}\right) \]

    12. metadata-eval 4.3%

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

13. Simplified 4.3%

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

14. Final simplification 4.3%

    \[\leadsto f \cdot \left(f \cdot \left(\pi \cdot \left(-0.08333333333333333\right)\right)\right) \]

Reproduce

herbie shell --seed 2023287 
(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)))))))))