VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.1% → 99.0%

Time: 38.6s

Alternatives: 4

Speedup: 3.3×

• could not determine a ground truth

  1. f = -2.369169900907371e+126

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 7.1% 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: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{{\tanh \left(\pi \cdot \left(f \cdot -0.25\right)\right)}^{-2}}\right) \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= (* (/ PI 4.0) f) 5e-43)
   (* (/ (- (log (/ 4.0 PI)) (log f)) PI) -4.0)
   (* (log (sqrt (pow (tanh (* PI (* f -0.25))) -2.0))) (/ -1.0 (/ PI 4.0)))))

C

double code(double f) {
	double tmp;
	if (((((double) M_PI) / 4.0) * f) <= 5e-43) {
		tmp = ((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI)) * -4.0;
	} else {
		tmp = log(sqrt(pow(tanh((((double) M_PI) * (f * -0.25))), -2.0))) * (-1.0 / (((double) M_PI) / 4.0));
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (((Math.PI / 4.0) * f) <= 5e-43) {
		tmp = ((Math.log((4.0 / Math.PI)) - Math.log(f)) / Math.PI) * -4.0;
	} else {
		tmp = Math.log(Math.sqrt(Math.pow(Math.tanh((Math.PI * (f * -0.25))), -2.0))) * (-1.0 / (Math.PI / 4.0));
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if ((math.pi / 4.0) * f) <= 5e-43:
		tmp = ((math.log((4.0 / math.pi)) - math.log(f)) / math.pi) * -4.0
	else:
		tmp = math.log(math.sqrt(math.pow(math.tanh((math.pi * (f * -0.25))), -2.0))) * (-1.0 / (math.pi / 4.0))
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (Float64(Float64(pi / 4.0) * f) <= 5e-43)
		tmp = Float64(Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi) * -4.0);
	else
		tmp = Float64(log(sqrt((tanh(Float64(pi * Float64(f * -0.25))) ^ -2.0))) * Float64(-1.0 / Float64(pi / 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (((pi / 4.0) * f) <= 5e-43)
		tmp = ((log((4.0 / pi)) - log(f)) / pi) * -4.0;
	else
		tmp = log(sqrt((tanh((pi * (f * -0.25))) ^ -2.0))) * (-1.0 / (pi / 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision], 5e-43], N[(N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[Log[N[Sqrt[N[Power[N[Tanh[N[(Pi * N[(f * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) 4) f) < 5.00000000000000019e-43

   1. Initial program 3.1%

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

      1. *-commutative 3.1%

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

      2. distribute-rgt-neg-in 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in f around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. mul-1-neg 99.5%

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

      3. unsub-neg 99.5%

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

      4. distribute-rgt-out-- 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-/r* 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in f around 0 99.5%

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

   if 5.00000000000000019e-43 < (*.f64 (/.f64 (PI.f64) 4) f)

   1. Initial program 24.9%

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

   3. Applied egg-rr 94.5%

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

      1. pow1/2 94.5%

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

      2. associate-*l* 94.5%

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

   5. Applied egg-rr 94.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left({\left({\left(\frac{1}{\tanh \left(\pi \cdot \left(-0.25 \cdot f\right)\right)}\right)}^{2}\right)}^{0.5}\right)} \]
   6. Step-by-step derivation

      1. unpow1/2 94.5%

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

      2. unpow2 94.5%

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

      3. unpow-1 94.5%

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

      4. unpow-1 94.5%

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

      5. pow-sqr 94.6%

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

      6. *-commutative 94.6%

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

      7. metadata-eval 94.6%

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

   7. Simplified 94.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\sqrt{{\tanh \left(\pi \cdot \left(f \cdot -0.25\right)\right)}^{-2}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{{\tanh \left(\pi \cdot \left(f \cdot -0.25\right)\right)}^{-2}}\right) \cdot \frac{-1}{\frac{\pi}{4}}\\ \end{array} \]

Alternative 2: 96.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (- (log (fma f (* PI 0.08333333333333333) (/ 2.0 (* PI (* f 0.5))))))
  (* PI 0.25)))

C

double code(double f) {
	return -log(fma(f, (((double) M_PI) * 0.08333333333333333), (2.0 / (((double) M_PI) * (f * 0.5))))) / (((double) M_PI) * 0.25);
}

Julia

function code(f)
	return Float64(Float64(-log(fma(f, Float64(pi * 0.08333333333333333), Float64(2.0 / Float64(pi * Float64(f * 0.5)))))) / Float64(pi * 0.25))
end

Wolfram

code[f_] := N[((-N[Log[N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(2.0 / N[(Pi * N[(f * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

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%

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

   1. *-un-lft-identity 95.7%

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

   2. fma-udef 95.7%

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

   3. div-inv 95.7%

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

   4. metadata-eval 95.7%

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

   5. associate-/r/ 95.7%

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

   6. unpow-prod-down 95.7%

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

   7. metadata-eval 95.7%

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

6. Applied egg-rr 95.7%

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

   1. expm1-log1p-u 95.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.0625 \cdot \left(\pi \cdot 2\right) + \left(\frac{{\pi}^{3}}{{\pi}^{2} \cdot 0.25} \cdot 0.005208333333333333\right) \cdot -2\right)\right)}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   2. expm1-udef 95.7%

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

   3. fma-def 95.7%

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

   4. associate-*l* 95.7%

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

   5. associate-/r* 95.7%

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

   6. pow-div 95.7%

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

   7. metadata-eval 95.7%

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

   8. pow1 95.7%

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

   9. metadata-eval 95.7%

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

8. Applied egg-rr 95.7%

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

   1. expm1-def 95.7%

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

   2. expm1-log1p 95.7%

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

   3. fma-udef 95.7%

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

   4. +-commutative 95.7%

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

   5. associate-*l/ 95.7%

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

   6. associate-/l* 95.7%

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

   7. metadata-eval 95.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \frac{\pi}{\color{blue}{-24}} + 0.0625 \cdot \left(\pi \cdot 2\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   8. *-commutative 95.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \frac{\pi}{-24} + \color{blue}{\left(\pi \cdot 2\right) \cdot 0.0625}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   9. associate-*l* 95.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \frac{\pi}{-24} + \color{blue}{\pi \cdot \left(2 \cdot 0.0625\right)}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   10. metadata-eval 95.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \frac{\pi}{-24} + \pi \cdot \color{blue}{0.125}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

10. Simplified 95.7%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\frac{\pi}{-24} + \pi \cdot 0.125}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]
11. Step-by-step derivation

    1. associate-*l/ 95.9%

       \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\mathsf{fma}\left(f, \frac{\pi}{-24} + \pi \cdot 0.125, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\frac{\pi}{4}}} \]

    2. *-un-lft-identity 95.9%

       \[\leadsto -\frac{\color{blue}{\log \left(\mathsf{fma}\left(f, \frac{\pi}{-24} + \pi \cdot 0.125, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}}{\frac{\pi}{4}} \]

    3. div-inv 95.9%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot \frac{1}{-24}} + \pi \cdot 0.125, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\frac{\pi}{4}} \]

    4. fma-def 95.9%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{fma}\left(\pi, \frac{1}{-24}, \pi \cdot 0.125\right)}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\frac{\pi}{4}} \]

    5. metadata-eval 95.9%

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

    6. div-inv 95.9%

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

12. Applied egg-rr 95.9%

    \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, -0.041666666666666664, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25}} \]
13. Step-by-step derivation

    1. fma-udef 95.9%

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

    2. distribute-lft-out 95.9%

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

    3. metadata-eval 95.9%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.08333333333333333}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25} \]

14. Simplified 95.9%

    \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25}} \]

15. Final simplification 95.9%

    \[\leadsto \frac{-\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25} \]

Alternative 3: 96.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{\frac{4}{f}}{\pi}\right)\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (- (/ (log (fma f (* PI 0.08333333333333333) (/ (/ 4.0 f) PI))) (* PI 0.25))))

C

double code(double f) {
	return -(log(fma(f, (((double) M_PI) * 0.08333333333333333), ((4.0 / f) / ((double) M_PI)))) / (((double) M_PI) * 0.25));
}

Julia

function code(f)
	return Float64(-Float64(log(fma(f, Float64(pi * 0.08333333333333333), Float64(Float64(4.0 / f) / pi))) / Float64(pi * 0.25)))
end

Wolfram

code[f_] := (-N[(N[Log[N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(N[(4.0 / f), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 6.3%

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

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%

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

   1. *-un-lft-identity 95.7%

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

   2. fma-udef 95.7%

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

   3. div-inv 95.7%

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

   4. metadata-eval 95.7%

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

   5. associate-/r/ 95.7%

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

   6. unpow-prod-down 95.7%

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

   7. metadata-eval 95.7%

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

6. Applied egg-rr 95.7%

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

   1. expm1-log1p-u 95.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.0625 \cdot \left(\pi \cdot 2\right) + \left(\frac{{\pi}^{3}}{{\pi}^{2} \cdot 0.25} \cdot 0.005208333333333333\right) \cdot -2\right)\right)}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   2. expm1-udef 95.7%

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

   3. fma-def 95.7%

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

   4. associate-*l* 95.7%

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

   5. associate-/r* 95.7%

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

   6. pow-div 95.7%

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

   7. metadata-eval 95.7%

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

   8. pow1 95.7%

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

   9. metadata-eval 95.7%

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

8. Applied egg-rr 95.7%

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

   1. expm1-def 95.7%

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

   2. expm1-log1p 95.7%

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

   3. fma-udef 95.7%

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

   4. +-commutative 95.7%

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

   5. associate-*l/ 95.7%

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

   6. associate-/l* 95.7%

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

   7. metadata-eval 95.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \frac{\pi}{\color{blue}{-24}} + 0.0625 \cdot \left(\pi \cdot 2\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   8. *-commutative 95.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \frac{\pi}{-24} + \color{blue}{\left(\pi \cdot 2\right) \cdot 0.0625}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   9. associate-*l* 95.7%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \frac{\pi}{-24} + \color{blue}{\pi \cdot \left(2 \cdot 0.0625\right)}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   10. metadata-eval 95.7%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \frac{\pi}{-24} + \pi \cdot \color{blue}{0.125}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

10. Simplified 95.7%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\frac{\pi}{-24} + \pi \cdot 0.125}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]
11. Step-by-step derivation

    1. associate-*l/ 95.9%

       \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\mathsf{fma}\left(f, \frac{\pi}{-24} + \pi \cdot 0.125, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\frac{\pi}{4}}} \]

    2. *-un-lft-identity 95.9%

       \[\leadsto -\frac{\color{blue}{\log \left(\mathsf{fma}\left(f, \frac{\pi}{-24} + \pi \cdot 0.125, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}}{\frac{\pi}{4}} \]

    3. div-inv 95.9%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot \frac{1}{-24}} + \pi \cdot 0.125, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\frac{\pi}{4}} \]

    4. fma-def 95.9%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{fma}\left(\pi, \frac{1}{-24}, \pi \cdot 0.125\right)}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\frac{\pi}{4}} \]

    5. metadata-eval 95.9%

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

    6. div-inv 95.9%

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

12. Applied egg-rr 95.9%

    \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\pi, -0.041666666666666664, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25}} \]
13. Step-by-step derivation

    1. fma-udef 95.9%

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

    2. distribute-lft-out 95.9%

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

    3. metadata-eval 95.9%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.08333333333333333}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25} \]

    4. *-commutative 95.9%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{2}{\color{blue}{\left(f \cdot 0.5\right) \cdot \pi}}\right)\right)}{\pi \cdot 0.25} \]

    5. associate-/r* 95.9%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \color{blue}{\frac{\frac{2}{f \cdot 0.5}}{\pi}}\right)\right)}{\pi \cdot 0.25} \]

    6. *-commutative 95.9%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{\frac{2}{\color{blue}{0.5 \cdot f}}}{\pi}\right)\right)}{\pi \cdot 0.25} \]

    7. associate-/r* 95.9%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{\color{blue}{\frac{\frac{2}{0.5}}{f}}}{\pi}\right)\right)}{\pi \cdot 0.25} \]

    8. metadata-eval 95.9%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{\frac{\color{blue}{4}}{f}}{\pi}\right)\right)}{\pi \cdot 0.25} \]

14. Simplified 95.9%

    \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{\frac{4}{f}}{\pi}\right)\right)}{\pi \cdot 0.25}} \]

15. Final simplification 95.9%

    \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{\frac{4}{f}}{\pi}\right)\right)}{\pi \cdot 0.25} \]

Alternative 4: 95.6% 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 6.3%

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

   1. *-commutative 6.3%

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

   2. distribute-rgt-neg-in 6.3%

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

3. Simplified 6.3%

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

4. Taylor expanded in f around 0 95.0%

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

   1. *-commutative 95.0%

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

   2. mul-1-neg 95.0%

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

   3. unsub-neg 95.0%

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

   4. distribute-rgt-out-- 95.0%

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

   5. metadata-eval 95.0%

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

   6. associate-/r* 95.0%

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

6. Simplified 95.0%

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

7. Taylor expanded in f around 0 95.0%

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

   1. metadata-eval 95.0%

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

   2. associate-*l/ 95.0%

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

   3. log-div 95.0%

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

   4. associate-*l/ 95.0%

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

   5. metadata-eval 95.0%

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

9. Simplified 95.0%

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

10. Final simplification 95.0%

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

Reproduce

herbie shell --seed 2023271 
(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)))))))))