VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.0% → 96.3%

Time: 39.0s

Alternatives: 6

Speedup: 3.3×

• could not determine a ground truth

  1. f = -4.602574880340157e+29

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: 7.0% 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, 1.4× speedup

Math

\[\begin{array}{l} \\ -\frac{\log \left(\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right) \cdot \frac{2}{f \cdot \left(\pi \cdot 0.5\right) + {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333}\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (/
   (log
    (*
     (cosh (* f (* PI 0.25)))
     (/ 2.0 (+ (* f (* PI 0.5)) (* (pow (* f PI) 3.0) 0.005208333333333333)))))
   (* PI 0.25))))

C

double code(double f) {
	return -(log((cosh((f * (((double) M_PI) * 0.25))) * (2.0 / ((f * (((double) M_PI) * 0.5)) + (pow((f * ((double) M_PI)), 3.0) * 0.005208333333333333))))) / (((double) M_PI) * 0.25));
}

Java

public static double code(double f) {
	return -(Math.log((Math.cosh((f * (Math.PI * 0.25))) * (2.0 / ((f * (Math.PI * 0.5)) + (Math.pow((f * Math.PI), 3.0) * 0.005208333333333333))))) / (Math.PI * 0.25));
}

Python

def code(f):
	return -(math.log((math.cosh((f * (math.pi * 0.25))) * (2.0 / ((f * (math.pi * 0.5)) + (math.pow((f * math.pi), 3.0) * 0.005208333333333333))))) / (math.pi * 0.25))

Julia

function code(f)
	return Float64(-Float64(log(Float64(cosh(Float64(f * Float64(pi * 0.25))) * Float64(2.0 / Float64(Float64(f * Float64(pi * 0.5)) + Float64((Float64(f * pi) ^ 3.0) * 0.005208333333333333))))) / Float64(pi * 0.25)))
end

MATLAB

function tmp = code(f)
	tmp = -(log((cosh((f * (pi * 0.25))) * (2.0 / ((f * (pi * 0.5)) + (((f * pi) ^ 3.0) * 0.005208333333333333))))) / (pi * 0.25));
end

Wolfram

code[f_] := (-N[(N[Log[N[(N[Cosh[N[(f * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 / N[(N[(f * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(f * Pi), $MachinePrecision], 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 7.6%

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

2. Taylor expanded in f around 0 98.5%

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

   1. fma-def 98.5%

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

   2. distribute-rgt-out-- 98.5%

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

   3. metadata-eval 98.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \]

   4. *-commutative 98.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) \cdot {f}^{3}}\right)}\right) \]

   5. distribute-rgt-out-- 98.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)} \cdot {f}^{3}\right)}\right) \]

   6. associate-*l* 98.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{\pi}^{3} \cdot \left(\left(0.0026041666666666665 - -0.0026041666666666665\right) \cdot {f}^{3}\right)}\right)}\right) \]

   7. metadata-eval 98.5%

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

4. Simplified 98.5%

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

   1. associate-*l/ 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. cosh-undef 98.6%

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

   4. div-inv 98.6%

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

   5. metadata-eval 98.6%

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

   6. div-inv 98.6%

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

   7. metadata-eval 98.6%

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

6. Applied egg-rr 98.6%

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

   1. expm1-log1p-u 98.6%

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

   2. expm1-udef 98.6%

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

   3. associate-/l* 98.6%

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

   4. associate-*l* 98.6%

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

8. Applied egg-rr 98.6%

   \[\leadsto -\frac{\log \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{\frac{\mathsf{fma}\left(f, \pi \cdot 0.5, {\pi}^{3} \cdot \left(0.005208333333333333 \cdot {f}^{3}\right)\right)}{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}}\right)} - 1\right)}}{\pi \cdot 0.25} \]
9. Step-by-step derivation

   1. expm1-def 98.6%

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

   2. expm1-log1p 98.6%

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

   3. associate-/r/ 98.6%

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

   4. *-commutative 98.6%

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

   5. associate-*r* 98.6%

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

   6. *-commutative 98.6%

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

   7. *-commutative 98.6%

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

   8. associate-*r* 98.6%

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

   9. *-commutative 98.6%

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

   10. cube-prod 98.6%

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

10. Simplified 98.6%

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

    1. fma-udef 98.6%

       \[\leadsto -\frac{\log \left(\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right) \cdot \frac{2}{\color{blue}{f \cdot \left(\pi \cdot 0.5\right) + {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333}}\right)}{\pi \cdot 0.25} \]

12. Applied egg-rr 98.6%

    \[\leadsto -\frac{\log \left(\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right) \cdot \frac{2}{\color{blue}{f \cdot \left(\pi \cdot 0.5\right) + {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333}}\right)}{\pi \cdot 0.25} \]

13. Final simplification 98.6%

    \[\leadsto -\frac{\log \left(\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right) \cdot \frac{2}{f \cdot \left(\pi \cdot 0.5\right) + {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333}\right)}{\pi \cdot 0.25} \]

Alternative 2: 96.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(\mathsf{fma}\left(2, f \cdot \left(\pi \cdot 0.041666666666666664\right), \frac{\frac{4}{f}}{\pi}\right)\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (- (log (fma 2.0 (* f (* PI 0.041666666666666664)) (/ (/ 4.0 f) PI))))
  (* PI 0.25)))

C

double code(double f) {
	return -log(fma(2.0, (f * (((double) M_PI) * 0.041666666666666664)), ((4.0 / f) / ((double) M_PI)))) / (((double) M_PI) * 0.25);
}

Julia

function code(f)
	return Float64(Float64(-log(fma(2.0, Float64(f * Float64(pi * 0.041666666666666664)), Float64(Float64(4.0 / f) / pi)))) / Float64(pi * 0.25))
end

Wolfram

code[f_] := N[((-N[Log[N[(2.0 * N[(f * N[(Pi * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 / f), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

2. Taylor expanded in f around 0 98.5%

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

   1. fma-def 98.5%

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

   2. distribute-rgt-out-- 98.5%

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

   3. metadata-eval 98.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \]

   4. *-commutative 98.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) \cdot {f}^{3}}\right)}\right) \]

   5. distribute-rgt-out-- 98.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)} \cdot {f}^{3}\right)}\right) \]

   6. associate-*l* 98.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{{\pi}^{3} \cdot \left(\left(0.0026041666666666665 - -0.0026041666666666665\right) \cdot {f}^{3}\right)}\right)}\right) \]

   7. metadata-eval 98.5%

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

4. Simplified 98.5%

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

   1. associate-*l/ 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. cosh-undef 98.6%

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

   4. div-inv 98.6%

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

   5. metadata-eval 98.6%

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

   6. div-inv 98.6%

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

   7. metadata-eval 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in f around 0 98.6%

   \[\leadsto -\frac{\log \color{blue}{\left(2 \cdot \left(f \cdot \left(0.0625 \cdot \pi - 0.020833333333333332 \cdot \pi\right)\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)}}{\pi \cdot 0.25} \]
8. Step-by-step derivation

   1. fma-def 98.6%

      \[\leadsto -\frac{\log \color{blue}{\left(\mathsf{fma}\left(2, f \cdot \left(0.0625 \cdot \pi - 0.020833333333333332 \cdot \pi\right), 4 \cdot \frac{1}{f \cdot \pi}\right)\right)}}{\pi \cdot 0.25} \]

   2. distribute-rgt-out-- 98.6%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(2, f \cdot \color{blue}{\left(\pi \cdot \left(0.0625 - 0.020833333333333332\right)\right)}, 4 \cdot \frac{1}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   3. metadata-eval 98.6%

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

   4. associate-*r/ 98.6%

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

   5. metadata-eval 98.6%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(2, f \cdot \left(\pi \cdot 0.041666666666666664\right), \frac{\color{blue}{4}}{f \cdot \pi}\right)\right)}{\pi \cdot 0.25} \]

   6. associate-/r* 98.6%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(2, f \cdot \left(\pi \cdot 0.041666666666666664\right), \color{blue}{\frac{\frac{4}{f}}{\pi}}\right)\right)}{\pi \cdot 0.25} \]

9. Simplified 98.6%

   \[\leadsto -\frac{\log \color{blue}{\left(\mathsf{fma}\left(2, f \cdot \left(\pi \cdot 0.041666666666666664\right), \frac{\frac{4}{f}}{\pi}\right)\right)}}{\pi \cdot 0.25} \]

10. Final simplification 98.6%

    \[\leadsto \frac{-\log \left(\mathsf{fma}\left(2, f \cdot \left(\pi \cdot 0.041666666666666664\right), \frac{\frac{4}{f}}{\pi}\right)\right)}{\pi \cdot 0.25} \]

Alternative 3: 95.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

   1. *-commutative 7.6%

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

   2. distribute-rgt-neg-in 7.6%

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

   3. associate-/r/ 7.6%

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

   4. associate-*l/ 7.6%

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

   5. metadata-eval 7.6%

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

   6. distribute-neg-frac 7.6%

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

3. Simplified 7.6%

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

4. Taylor expanded in f around 0 97.9%

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

   1. *-commutative 97.9%

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

   2. mul-1-neg 97.9%

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

   3. unsub-neg 97.9%

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

   4. distribute-rgt-out-- 97.9%

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

   5. *-commutative 97.9%

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

   6. associate-/r* 97.9%

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

   7. metadata-eval 97.9%

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

   8. metadata-eval 97.9%

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

6. Simplified 97.9%

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

7. Final simplification 97.9%

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

Alternative 4: 95.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

   1. *-commutative 7.6%

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

   2. distribute-rgt-neg-in 7.6%

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

   3. associate-/r/ 7.6%

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

   4. associate-*l/ 7.6%

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

   5. metadata-eval 7.6%

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

   6. distribute-neg-frac 7.6%

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

3. Simplified 7.6%

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

4. Taylor expanded in f around 0 97.9%

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

   1. associate-*r/ 97.9%

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

   2. associate-/l* 97.7%

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

   3. mul-1-neg 97.7%

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

   4. unsub-neg 97.7%

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

   5. distribute-rgt-out-- 97.7%

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

   6. metadata-eval 97.7%

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

6. Simplified 97.7%

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

7. Taylor expanded in f around 0 97.9%

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

   1. associate-*r/ 97.9%

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

   2. sub-neg 97.9%

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

   3. log-rec 97.9%

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

   4. distribute-rgt-in 97.9%

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

   5. metadata-eval 97.9%

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

   6. associate-/r* 97.9%

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

   7. *-commutative 97.9%

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

   8. distribute-rgt-in 97.9%

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

   9. log-rec 97.9%

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

   10. unsub-neg 97.9%

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

   11. log-div 97.8%

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

   12. associate-*l/ 97.7%

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

9. Simplified 97.7%

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

10. Final simplification 97.7%

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

Alternative 5: 95.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

   1. *-commutative 7.6%

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

   2. distribute-rgt-neg-in 7.6%

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

   3. associate-/r/ 7.6%

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

   4. associate-*l/ 7.6%

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

   5. metadata-eval 7.6%

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

   6. distribute-neg-frac 7.6%

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

3. Simplified 7.6%

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

4. Taylor expanded in f around 0 97.9%

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

   1. *-commutative 97.9%

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

   2. mul-1-neg 97.9%

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

   3. unsub-neg 97.9%

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

   4. distribute-rgt-out-- 97.9%

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

   5. *-commutative 97.9%

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

   6. associate-/r* 97.9%

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

   7. metadata-eval 97.9%

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

   8. metadata-eval 97.9%

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

6. Simplified 97.9%

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

7. Taylor expanded in f around inf 97.9%

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

   1. div-sub 97.8%

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

   2. mul-1-neg 97.8%

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

   3. log-rec 97.8%

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

   4. remove-double-neg 97.8%

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

   5. div-sub 97.9%

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

   6. log-div 97.9%

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

   7. associate--l- 97.8%

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

   8. log-prod 97.8%

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

   9. *-commutative 97.8%

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

   10. log-div 97.8%

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

   11. *-commutative 97.8%

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

9. Simplified 97.8%

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

10. Final simplification 97.8%

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

Alternative 6: 95.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

   1. *-commutative 7.6%

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

   2. distribute-rgt-neg-in 7.6%

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

   3. associate-/r/ 7.6%

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

   4. associate-*l/ 7.6%

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

   5. metadata-eval 7.6%

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

   6. distribute-neg-frac 7.6%

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

3. Simplified 7.6%

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

4. Taylor expanded in f around 0 97.9%

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

   1. *-commutative 97.9%

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

   2. mul-1-neg 97.9%

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

   3. unsub-neg 97.9%

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

   4. distribute-rgt-out-- 97.9%

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

   5. *-commutative 97.9%

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

   6. associate-/r* 97.9%

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

   7. metadata-eval 97.9%

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

   8. metadata-eval 97.9%

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

6. Simplified 97.9%

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

   1. add-exp-log 96.6%

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

   2. diff-log 96.6%

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

   3. associate-/l/ 96.6%

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

8. Applied egg-rr 96.6%

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

   1. add-exp-log 97.8%

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

   2. *-commutative 97.8%

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

   3. associate-/r* 97.8%

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

10. Applied egg-rr 97.8%

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

11. Final simplification 97.8%

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

Reproduce

herbie shell --seed 2023282 
(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)))))))))