VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 96.3%

Time: 33.9s

Alternatives: 9

Speedup: 3.3×

• could not determine a ground truth

  1. f = -3.9645528121010563e+121

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: 6.6% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{-4 \cdot \log \left(\frac{{\left(e^{\frac{\pi}{-4}}\right)}^{f} + {\left(e^{\pi}\right)}^{\left(f \cdot 0.25\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, {\left(\pi \cdot f\right)}^{3} \cdot 0.005208333333333333\right)}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (*
   -4.0
   (log
    (/
     (+ (pow (exp (/ PI -4.0)) f) (pow (exp PI) (* f 0.25)))
     (fma f (* PI 0.5) (* (pow (* PI f) 3.0) 0.005208333333333333)))))
  PI))

C

double code(double f) {
	return (-4.0 * log(((pow(exp((((double) M_PI) / -4.0)), f) + pow(exp(((double) M_PI)), (f * 0.25))) / fma(f, (((double) M_PI) * 0.5), (pow((((double) M_PI) * f), 3.0) * 0.005208333333333333))))) / ((double) M_PI);
}

Julia

function code(f)
	return Float64(Float64(-4.0 * log(Float64(Float64((exp(Float64(pi / -4.0)) ^ f) + (exp(pi) ^ Float64(f * 0.25))) / fma(f, Float64(pi * 0.5), Float64((Float64(pi * f) ^ 3.0) * 0.005208333333333333))))) / pi)
end

Wolfram

code[f_] := N[(N[(-4.0 * N[Log[N[(N[(N[Power[N[Exp[N[(Pi / -4.0), $MachinePrecision]], $MachinePrecision], f], $MachinePrecision] + N[Power[N[Exp[Pi], $MachinePrecision], N[(f * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(f * N[(Pi * 0.5), $MachinePrecision] + N[(N[Power[N[(Pi * f), $MachinePrecision], 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 7.8%

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

   1. distribute-lft-neg-in 7.8%

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

   2. *-commutative 7.8%

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

3. Simplified 7.8%

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

4. Taylor expanded in f around 0 98.0%

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

   1. fma-def 98.0%

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

   2. distribute-rgt-out-- 98.0%

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

   3. metadata-eval 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   4. distribute-rgt-out-- 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, {f}^{3} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   5. associate-*r* 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{3} \cdot {\pi}^{3}\right) \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   6. cube-prod 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{\left(f \cdot \pi\right)}^{3}} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   7. metadata-eval 98.0%

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

6. Simplified 98.0%

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

   1. associate-*r/ 98.2%

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

8. Applied egg-rr 98.2%

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

9. Final simplification 98.2%

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

Alternative 2: 96.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, {\left(\sqrt[3]{\frac{\frac{4}{f}}{\pi}}\right)}^{3}\right)\right)}{\frac{\pi}{-4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (log (fma f (* PI 0.08333333333333333) (pow (cbrt (/ (/ 4.0 f) PI)) 3.0)))
  (/ PI -4.0)))

C

double code(double f) {
	return log(fma(f, (((double) M_PI) * 0.08333333333333333), pow(cbrt(((4.0 / f) / ((double) M_PI))), 3.0))) / (((double) M_PI) / -4.0);
}

Julia

function code(f)
	return Float64(log(fma(f, Float64(pi * 0.08333333333333333), (cbrt(Float64(Float64(4.0 / f) / pi)) ^ 3.0))) / Float64(pi / -4.0))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision] + N[Power[N[Power[N[(N[(4.0 / f), $MachinePrecision] / Pi), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi / -4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.8%

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

   1. distribute-lft-neg-in 7.8%

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

   2. *-commutative 7.8%

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

3. Simplified 7.8%

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

4. Taylor expanded in f around 0 98.0%

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

   1. fma-def 98.0%

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

   2. distribute-rgt-out-- 98.0%

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

   3. metadata-eval 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   4. distribute-rgt-out-- 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, {f}^{3} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   5. associate-*r* 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{3} \cdot {\pi}^{3}\right) \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   6. cube-prod 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{\left(f \cdot \pi\right)}^{3}} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   7. metadata-eval 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in f around 0 98.0%

   \[\leadsto \log \color{blue}{\left(2 \cdot \frac{-0.25 \cdot \pi + 0.25 \cdot \pi}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right)} \cdot \frac{-4}{\pi} \]
8. Step-by-step derivation

   1. associate-*r/ 98.0%

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

   2. distribute-rgt-out 98.0%

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

   3. metadata-eval 98.0%

      \[\leadsto \log \left(\frac{2 \cdot \left(\pi \cdot \color{blue}{0}\right)}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   4. mul0-rgt 98.0%

      \[\leadsto \log \left(\frac{2 \cdot \color{blue}{0}}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   5. mul0-rgt 98.0%

      \[\leadsto \log \left(\frac{2 \cdot \color{blue}{\left({\pi}^{3} \cdot 0\right)}}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   6. metadata-eval 98.0%

      \[\leadsto \log \left(\frac{2 \cdot \left({\pi}^{3} \cdot \color{blue}{\left(-0.0026041666666666665 + 0.0026041666666666665\right)}\right)}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   7. distribute-rgt-out 98.0%

      \[\leadsto \log \left(\frac{2 \cdot \color{blue}{\left(-0.0026041666666666665 \cdot {\pi}^{3} + 0.0026041666666666665 \cdot {\pi}^{3}\right)}}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   8. distribute-rgt-out 98.0%

      \[\leadsto \log \left(\frac{2 \cdot \color{blue}{\left({\pi}^{3} \cdot \left(-0.0026041666666666665 + 0.0026041666666666665\right)\right)}}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   9. metadata-eval 98.0%

      \[\leadsto \log \left(\frac{2 \cdot \left({\pi}^{3} \cdot \color{blue}{0}\right)}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   10. mul0-rgt 98.0%

       \[\leadsto \log \left(\frac{2 \cdot \color{blue}{0}}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   11. metadata-eval 98.0%

       \[\leadsto \log \left(\frac{\color{blue}{0}}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   12. fma-def 98.0%

       \[\leadsto \log \left(\frac{0}{\pi} + \color{blue}{\mathsf{fma}\left(f, 0.125 \cdot \pi - 0.041666666666666664 \cdot \pi, 4 \cdot \frac{1}{f \cdot \pi}\right)}\right) \cdot \frac{-4}{\pi} \]

   13. distribute-rgt-out-- 98.0%

       \[\leadsto \log \left(\frac{0}{\pi} + \mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.125 - 0.041666666666666664\right)}, 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   14. metadata-eval 98.0%

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

9. Simplified 98.0%

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

    1. associate-*r/ 98.1%

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

    2. div0 98.1%

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

    3. associate-/r* 98.1%

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

11. Applied egg-rr 98.1%

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

    1. associate-/l* 98.1%

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

    2. +-lft-identity 98.1%

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

13. Simplified 98.1%

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

    1. add-cube-cbrt 98.1%

       \[\leadsto \frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \color{blue}{\left(\sqrt[3]{\frac{\frac{4}{\pi}}{f}} \cdot \sqrt[3]{\frac{\frac{4}{\pi}}{f}}\right) \cdot \sqrt[3]{\frac{\frac{4}{\pi}}{f}}}\right)\right)}{\frac{\pi}{-4}} \]

    2. pow3 98.1%

       \[\leadsto \frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \color{blue}{{\left(\sqrt[3]{\frac{\frac{4}{\pi}}{f}}\right)}^{3}}\right)\right)}{\frac{\pi}{-4}} \]

    3. associate-/l/ 98.1%

       \[\leadsto \frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, {\left(\sqrt[3]{\color{blue}{\frac{4}{f \cdot \pi}}}\right)}^{3}\right)\right)}{\frac{\pi}{-4}} \]

    4. associate-/r* 98.1%

       \[\leadsto \frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, {\left(\sqrt[3]{\color{blue}{\frac{\frac{4}{f}}{\pi}}}\right)}^{3}\right)\right)}{\frac{\pi}{-4}} \]

15. Applied egg-rr 98.1%

    \[\leadsto \frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \color{blue}{{\left(\sqrt[3]{\frac{\frac{4}{f}}{\pi}}\right)}^{3}}\right)\right)}{\frac{\pi}{-4}} \]

16. Final simplification 98.1%

    \[\leadsto \frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, {\left(\sqrt[3]{\frac{\frac{4}{f}}{\pi}}\right)}^{3}\right)\right)}{\frac{\pi}{-4}} \]

Alternative 3: 96.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.8%

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

   1. distribute-lft-neg-in 7.8%

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

   2. *-commutative 7.8%

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

3. Simplified 7.8%

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

4. Taylor expanded in f around 0 98.0%

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

   1. fma-def 98.0%

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

   2. distribute-rgt-out-- 98.0%

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

   3. metadata-eval 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   4. distribute-rgt-out-- 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, {f}^{3} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   5. associate-*r* 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{3} \cdot {\pi}^{3}\right) \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   6. cube-prod 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{\left(f \cdot \pi\right)}^{3}} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   7. metadata-eval 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in f around 0 98.0%

   \[\leadsto \log \color{blue}{\left(2 \cdot \frac{-0.25 \cdot \pi + 0.25 \cdot \pi}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right)} \cdot \frac{-4}{\pi} \]
8. Step-by-step derivation

   1. associate-*r/ 98.0%

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

   2. distribute-rgt-out 98.0%

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

   3. metadata-eval 98.0%

      \[\leadsto \log \left(\frac{2 \cdot \left(\pi \cdot \color{blue}{0}\right)}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   4. mul0-rgt 98.0%

      \[\leadsto \log \left(\frac{2 \cdot \color{blue}{0}}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   5. mul0-rgt 98.0%

      \[\leadsto \log \left(\frac{2 \cdot \color{blue}{\left({\pi}^{3} \cdot 0\right)}}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   6. metadata-eval 98.0%

      \[\leadsto \log \left(\frac{2 \cdot \left({\pi}^{3} \cdot \color{blue}{\left(-0.0026041666666666665 + 0.0026041666666666665\right)}\right)}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   7. distribute-rgt-out 98.0%

      \[\leadsto \log \left(\frac{2 \cdot \color{blue}{\left(-0.0026041666666666665 \cdot {\pi}^{3} + 0.0026041666666666665 \cdot {\pi}^{3}\right)}}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   8. distribute-rgt-out 98.0%

      \[\leadsto \log \left(\frac{2 \cdot \color{blue}{\left({\pi}^{3} \cdot \left(-0.0026041666666666665 + 0.0026041666666666665\right)\right)}}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   9. metadata-eval 98.0%

      \[\leadsto \log \left(\frac{2 \cdot \left({\pi}^{3} \cdot \color{blue}{0}\right)}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   10. mul0-rgt 98.0%

       \[\leadsto \log \left(\frac{2 \cdot \color{blue}{0}}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   11. metadata-eval 98.0%

       \[\leadsto \log \left(\frac{\color{blue}{0}}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   12. fma-def 98.0%

       \[\leadsto \log \left(\frac{0}{\pi} + \color{blue}{\mathsf{fma}\left(f, 0.125 \cdot \pi - 0.041666666666666664 \cdot \pi, 4 \cdot \frac{1}{f \cdot \pi}\right)}\right) \cdot \frac{-4}{\pi} \]

   13. distribute-rgt-out-- 98.0%

       \[\leadsto \log \left(\frac{0}{\pi} + \mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.125 - 0.041666666666666664\right)}, 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot \frac{-4}{\pi} \]

   14. metadata-eval 98.0%

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

9. Simplified 98.0%

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

    1. associate-*r/ 98.1%

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

    2. div0 98.1%

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

    3. associate-/r* 98.1%

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

11. Applied egg-rr 98.1%

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

    1. associate-/l* 98.1%

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

    2. +-lft-identity 98.1%

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

13. Simplified 98.1%

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

14. Final simplification 98.1%

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

Alternative 4: 95.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \log \left(0.125 \cdot \left(\pi \cdot f\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) \cdot \frac{-4}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (log (+ (* 0.125 (* PI f)) (* 4.0 (/ 1.0 (* PI f))))) (/ -4.0 PI)))

C

double code(double f) {
	return log(((0.125 * (((double) M_PI) * f)) + (4.0 * (1.0 / (((double) M_PI) * f))))) * (-4.0 / ((double) M_PI));
}

Java

public static double code(double f) {
	return Math.log(((0.125 * (Math.PI * f)) + (4.0 * (1.0 / (Math.PI * f))))) * (-4.0 / Math.PI);
}

Python

def code(f):
	return math.log(((0.125 * (math.pi * f)) + (4.0 * (1.0 / (math.pi * f))))) * (-4.0 / math.pi)

Julia

function code(f)
	return Float64(log(Float64(Float64(0.125 * Float64(pi * f)) + Float64(4.0 * Float64(1.0 / Float64(pi * f))))) * Float64(-4.0 / pi))
end

MATLAB

function tmp = code(f)
	tmp = log(((0.125 * (pi * f)) + (4.0 * (1.0 / (pi * f))))) * (-4.0 / pi);
end

Wolfram

code[f_] := N[(N[Log[N[(N[(0.125 * N[(Pi * f), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(1.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.8%

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

   1. distribute-lft-neg-in 7.8%

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

   2. *-commutative 7.8%

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

3. Simplified 7.8%

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

4. Taylor expanded in f around 0 97.4%

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

   1. distribute-rgt-out-- 97.4%

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

   2. metadata-eval 97.4%

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

6. Simplified 97.4%

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

7. Taylor expanded in f around 0 97.4%

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

8. Final simplification 97.4%

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

Alternative 5: 95.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.8%

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

   1. distribute-lft-neg-in 7.8%

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

   2. *-commutative 7.8%

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

3. Simplified 7.8%

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

4. Taylor expanded in f around 0 97.4%

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

   1. distribute-rgt-out-- 97.4%

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

   2. metadata-eval 97.4%

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

6. Simplified 97.4%

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

7. Taylor expanded in f around 0 97.4%

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

   1. +-commutative 97.4%

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

   2. fma-def 97.4%

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

   3. associate-/r* 97.4%

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

   4. *-commutative 97.4%

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

   5. *-commutative 97.4%

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

9. Applied egg-rr 97.4%

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

    1. fma-udef 97.4%

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

    2. associate-/r* 97.4%

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

    3. associate-*r/ 97.4%

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

    4. metadata-eval 97.4%

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

    5. associate-/l/ 97.4%

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

    6. associate-*l* 97.4%

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

11. Simplified 97.4%

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

12. Final simplification 97.4%

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

Alternative 6: 95.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.8%

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

   1. distribute-lft-neg-in 7.8%

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

   2. *-commutative 7.8%

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

3. Simplified 7.8%

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

4. Taylor expanded in f around 0 97.4%

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

   1. mul-1-neg 97.4%

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

   2. unsub-neg 97.4%

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

   3. distribute-rgt-out-- 97.4%

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

   4. metadata-eval 97.4%

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

   5. metadata-eval 97.4%

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

   6. associate-/r* 97.4%

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

   7. metadata-eval 97.4%

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

6. Simplified 97.4%

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

7. Final simplification 97.4%

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

Alternative 7: 1.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.8%

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

   1. distribute-lft-neg-in 7.8%

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

   2. *-commutative 7.8%

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

3. Simplified 7.8%

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

4. Taylor expanded in f around 0 97.4%

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

   1. distribute-rgt-out-- 97.4%

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

   2. metadata-eval 97.4%

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

6. Simplified 97.4%

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

7. Taylor expanded in f around 0 97.4%

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

8. Taylor expanded in f around inf 1.6%

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

   1. *-commutative 1.6%

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

   2. *-commutative 1.6%

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

   3. associate-*l* 1.6%

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

10. Simplified 1.6%

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

11. Final simplification 1.6%

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

Alternative 8: 95.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.8%

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

   1. distribute-lft-neg-in 7.8%

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

   2. *-commutative 7.8%

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

3. Simplified 7.8%

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

4. Taylor expanded in f around 0 98.0%

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

   1. fma-def 98.0%

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

   2. distribute-rgt-out-- 98.0%

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

   3. metadata-eval 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   4. distribute-rgt-out-- 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, {f}^{3} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   5. associate-*r* 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{3} \cdot {\pi}^{3}\right) \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   6. cube-prod 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{\left(f \cdot \pi\right)}^{3}} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   7. metadata-eval 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in f around 0 97.2%

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

   1. *-commutative 97.2%

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

9. Simplified 97.2%

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

10. Final simplification 97.2%

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

Alternative 9: 95.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.8%

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

   1. distribute-lft-neg-in 7.8%

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

   2. *-commutative 7.8%

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

3. Simplified 7.8%

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

4. Taylor expanded in f around 0 98.0%

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

   1. fma-def 98.0%

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

   2. distribute-rgt-out-- 98.0%

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

   3. metadata-eval 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   4. distribute-rgt-out-- 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, {f}^{3} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   5. associate-*r* 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{3} \cdot {\pi}^{3}\right) \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   6. cube-prod 98.0%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{\left(f \cdot \pi\right)}^{3}} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   7. metadata-eval 98.0%

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

6. Simplified 98.0%

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

   1. associate-*r/ 98.2%

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

8. Applied egg-rr 98.2%

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

9. Taylor expanded in f around 0 97.4%

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

    1. *-commutative 97.4%

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

11. Simplified 97.4%

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

12. Final simplification 97.4%

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

Reproduce

herbie shell --seed 2023321 
(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)))))))))