VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 96.4%

Time: 29.9s

Alternatives: 9

Speedup: 3.3×

• could not determine a ground truth

  1. f = -1.0942405972267847e+204

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: 6.9% 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: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ -\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{\pi}{0.5}, \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}\right), 0\right)}{\pi}, f \cdot f, 0\right), \frac{\left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right) \cdot 4}{\pi}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (fma
   2.0
   (fma
    (/
     (fma
      (* PI 0.5)
      (fma 0.0625 (/ PI 0.5) (/ -2.0 (* (/ (pow PI 2.0) (pow PI 3.0)) 48.0)))
      0.0)
     PI)
    (* f f)
    0.0)
   (/ (* (- (log (/ 2.0 (* PI 0.5))) (log f)) 4.0) PI))))

C

double code(double f) {
	return -fma(2.0, fma((fma((((double) M_PI) * 0.5), fma(0.0625, (((double) M_PI) / 0.5), (-2.0 / ((pow(((double) M_PI), 2.0) / pow(((double) M_PI), 3.0)) * 48.0))), 0.0) / ((double) M_PI)), (f * f), 0.0), (((log((2.0 / (((double) M_PI) * 0.5))) - log(f)) * 4.0) / ((double) M_PI)));
}

Julia

function code(f)
	return Float64(-fma(2.0, fma(Float64(fma(Float64(pi * 0.5), fma(0.0625, Float64(pi / 0.5), Float64(-2.0 / Float64(Float64((pi ^ 2.0) / (pi ^ 3.0)) * 48.0))), 0.0) / pi), Float64(f * f), 0.0), Float64(Float64(Float64(log(Float64(2.0 / Float64(pi * 0.5))) - log(f)) * 4.0) / pi)))
end

Wolfram

code[f_] := (-N[(2.0 * N[(N[(N[(N[(Pi * 0.5), $MachinePrecision] * N[(0.0625 * N[(Pi / 0.5), $MachinePrecision] + N[(-2.0 / N[(N[(N[Power[Pi, 2.0], $MachinePrecision] / N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] * 48.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision] / Pi), $MachinePrecision] * N[(f * f), $MachinePrecision] + 0.0), $MachinePrecision] + N[(N[(N[(N[Log[N[(2.0 / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in f around 0 96.7%

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

4. Simplified 96.7%

   \[\leadsto -\color{blue}{\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{1 \cdot \pi}{0.5}, \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}\right), 0\right)}{\pi}, f \cdot f, 0\right), \frac{\left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right) \cdot 4}{\pi}\right)} \]

5. Final simplification 96.7%

   \[\leadsto -\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{\pi}{0.5}, \frac{-2}{\frac{{\pi}^{2}}{{\pi}^{3}} \cdot 48}\right), 0\right)}{\pi}, f \cdot f, 0\right), \frac{\left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right) \cdot 4}{\pi}\right) \]

Alternative 2: 96.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := f \cdot \left(\pi \cdot 0.25\right)\\ t_1 := \sqrt[3]{\cosh t_0}\\ \frac{-\log \left({t_1}^{2} \cdot \frac{t_1}{\sinh t_0}\right)}{\pi \cdot 0.25} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* f (* PI 0.25))) (t_1 (cbrt (cosh t_0))))
   (/ (- (log (* (pow t_1 2.0) (/ t_1 (sinh t_0))))) (* PI 0.25))))

C

double code(double f) {
	double t_0 = f * (((double) M_PI) * 0.25);
	double t_1 = cbrt(cosh(t_0));
	return -log((pow(t_1, 2.0) * (t_1 / sinh(t_0)))) / (((double) M_PI) * 0.25);
}

Java

public static double code(double f) {
	double t_0 = f * (Math.PI * 0.25);
	double t_1 = Math.cbrt(Math.cosh(t_0));
	return -Math.log((Math.pow(t_1, 2.0) * (t_1 / Math.sinh(t_0)))) / (Math.PI * 0.25);
}

Julia

function code(f)
	t_0 = Float64(f * Float64(pi * 0.25))
	t_1 = cbrt(cosh(t_0))
	return Float64(Float64(-log(Float64((t_1 ^ 2.0) * Float64(t_1 / sinh(t_0))))) / Float64(pi * 0.25))
end

Wolfram

code[f_] := Block[{t$95$0 = N[(f * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cosh[t$95$0], $MachinePrecision], 1/3], $MachinePrecision]}, N[((-N[Log[N[(N[Power[t$95$1, 2.0], $MachinePrecision] * N[(t$95$1 / N[Sinh[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in f around inf 7.0%

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

   1. exp-prod 7.0%

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

   2. *-commutative 7.0%

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

   3. distribute-lft-neg-in 7.0%

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

   4. metadata-eval 7.0%

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

   5. *-commutative 7.0%

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

4. Simplified 7.0%

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

   1. associate-*l/ 7.0%

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

6. Applied egg-rr 96.6%

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

8. Simplified 96.6%

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

   1. add-exp-log 96.6%

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

10. Applied egg-rr 96.6%

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

    1. add-cube-cbrt 96.6%

       \[\leadsto -\frac{\log \left(\frac{\color{blue}{\left(\sqrt[3]{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)} \cdot \sqrt[3]{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}\right) \cdot \sqrt[3]{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}}}{\sinh \left(e^{\log \left(f \cdot \left(\pi \cdot 0.25\right)\right)}\right)}\right)}{\pi \cdot 0.25} \]

    2. add-exp-log 96.7%

       \[\leadsto -\frac{\log \left(\frac{\left(\sqrt[3]{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)} \cdot \sqrt[3]{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}\right) \cdot \sqrt[3]{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}}{\sinh \color{blue}{\left(f \cdot \left(\pi \cdot 0.25\right)\right)}}\right)}{\pi \cdot 0.25} \]

    3. *-un-lft-identity 96.7%

       \[\leadsto -\frac{\log \left(\frac{\left(\sqrt[3]{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)} \cdot \sqrt[3]{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}\right) \cdot \sqrt[3]{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}}{\color{blue}{1 \cdot \sinh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}}\right)}{\pi \cdot 0.25} \]

    4. times-frac 96.7%

       \[\leadsto -\frac{\log \color{blue}{\left(\frac{\sqrt[3]{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)} \cdot \sqrt[3]{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}}{1} \cdot \frac{\sqrt[3]{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}}{\sinh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}\right)}}{\pi \cdot 0.25} \]

    5. pow2 96.7%

       \[\leadsto -\frac{\log \left(\frac{\color{blue}{{\left(\sqrt[3]{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}}{\sinh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}\right)}{\pi \cdot 0.25} \]

12. Applied egg-rr 96.7%

    \[\leadsto -\frac{\log \color{blue}{\left(\frac{{\left(\sqrt[3]{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}}{\sinh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}\right)}}{\pi \cdot 0.25} \]

13. Final simplification 96.7%

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

Alternative 3: 96.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot \left(\pi \cdot f\right)\\ \log \left(\frac{\cosh t_0}{\sinh t_0}\right) \cdot \frac{-4}{\pi} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* 0.25 (* PI f))))
   (* (log (/ (cosh t_0) (sinh t_0))) (/ (- 4.0) PI))))

C

double code(double f) {
	double t_0 = 0.25 * (((double) M_PI) * f);
	return log((cosh(t_0) / sinh(t_0))) * (-4.0 / ((double) M_PI));
}

Java

public static double code(double f) {
	double t_0 = 0.25 * (Math.PI * f);
	return Math.log((Math.cosh(t_0) / Math.sinh(t_0))) * (-4.0 / Math.PI);
}

Python

def code(f):
	t_0 = 0.25 * (math.pi * f)
	return math.log((math.cosh(t_0) / math.sinh(t_0))) * (-4.0 / math.pi)

Julia

function code(f)
	t_0 = Float64(0.25 * Float64(pi * f))
	return Float64(log(Float64(cosh(t_0) / sinh(t_0))) * Float64(Float64(-4.0) / pi))
end

MATLAB

function tmp = code(f)
	t_0 = 0.25 * (pi * f);
	tmp = log((cosh(t_0) / sinh(t_0))) * (-4.0 / pi);
end

Wolfram

code[f_] := Block[{t$95$0 = N[(0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]}, N[(N[Log[N[(N[Cosh[t$95$0], $MachinePrecision] / N[Sinh[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-4.0) / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in f around inf 7.0%

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

   1. exp-prod 7.0%

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

   2. *-commutative 7.0%

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

   3. distribute-lft-neg-in 7.0%

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

   4. metadata-eval 7.0%

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

   5. *-commutative 7.0%

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

4. Simplified 7.0%

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

   1. associate-*l/ 7.0%

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

6. Applied egg-rr 96.6%

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

8. Simplified 96.6%

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

   1. add-exp-log 96.6%

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

10. Applied egg-rr 96.6%

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

    1. expm1-log1p-u 95.4%

       \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\sinh \left(e^{\log \left(f \cdot \left(\pi \cdot 0.25\right)\right)}\right)}\right)}{\pi \cdot 0.25}\right)\right)} \]

    2. expm1-udef 95.4%

       \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \left(\frac{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\sinh \left(e^{\log \left(f \cdot \left(\pi \cdot 0.25\right)\right)}\right)}\right)}{\pi \cdot 0.25}\right)} - 1\right)} \]

12. Applied egg-rr 95.4%

    \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(4 \cdot \frac{\log \left(\frac{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\sinh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}\right)}{\pi}\right)} - 1\right)} \]
13. Step-by-step derivation

    1. expm1-def 95.4%

       \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(4 \cdot \frac{\log \left(\frac{\cosh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{\sinh \left(f \cdot \left(\pi \cdot 0.25\right)\right)}\right)}{\pi}\right)\right)} \]

    2. expm1-log1p 96.6%

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

    3. *-commutative 96.6%

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

    4. associate-*l/ 96.6%

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

    5. associate-*r/ 96.4%

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

    6. associate-*r* 96.4%

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

    7. *-commutative 96.4%

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

    8. associate-*r* 96.4%

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

    9. *-commutative 96.4%

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

14. Simplified 96.4%

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

15. Final simplification 96.4%

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

Alternative 4: 97.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := f \cdot \left(\pi \cdot 0.25\right)\\ \frac{-\log \left(\frac{\cosh t_0}{\sinh t_0}\right)}{\pi \cdot 0.25} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* f (* PI 0.25))))
   (/ (- (log (/ (cosh t_0) (sinh t_0)))) (* PI 0.25))))

C

double code(double f) {
	double t_0 = f * (((double) M_PI) * 0.25);
	return -log((cosh(t_0) / sinh(t_0))) / (((double) M_PI) * 0.25);
}

Java

public static double code(double f) {
	double t_0 = f * (Math.PI * 0.25);
	return -Math.log((Math.cosh(t_0) / Math.sinh(t_0))) / (Math.PI * 0.25);
}

Python

def code(f):
	t_0 = f * (math.pi * 0.25)
	return -math.log((math.cosh(t_0) / math.sinh(t_0))) / (math.pi * 0.25)

Julia

function code(f)
	t_0 = Float64(f * Float64(pi * 0.25))
	return Float64(Float64(-log(Float64(cosh(t_0) / sinh(t_0)))) / Float64(pi * 0.25))
end

MATLAB

function tmp = code(f)
	t_0 = f * (pi * 0.25);
	tmp = -log((cosh(t_0) / sinh(t_0))) / (pi * 0.25);
end

Wolfram

code[f_] := Block[{t$95$0 = N[(f * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]}, N[((-N[Log[N[(N[Cosh[t$95$0], $MachinePrecision] / N[Sinh[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in f around inf 7.0%

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

   1. exp-prod 7.0%

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

   2. *-commutative 7.0%

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

   3. distribute-lft-neg-in 7.0%

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

   4. metadata-eval 7.0%

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

   5. *-commutative 7.0%

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

4. Simplified 7.0%

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

   1. associate-*l/ 7.0%

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

6. Applied egg-rr 96.6%

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

8. Simplified 96.6%

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

9. Final simplification 96.6%

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

Alternative 5: 95.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in f around inf 7.0%

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

   1. exp-prod 7.0%

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

   2. *-commutative 7.0%

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

   3. distribute-lft-neg-in 7.0%

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

   4. metadata-eval 7.0%

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

   5. *-commutative 7.0%

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

4. Simplified 7.0%

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

   1. associate-*l/ 7.0%

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

6. Applied egg-rr 96.6%

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

8. Simplified 96.6%

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

9. Taylor expanded in f around 0 95.7%

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

    1. associate-/r* 95.7%

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

11. Simplified 95.7%

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

    1. add-sqr-sqrt 95.1%

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

    2. sqrt-unprod 95.8%

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

    3. pow2 95.8%

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

    4. div-inv 95.8%

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

    5. frac-times 95.8%

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

    6. metadata-eval 95.8%

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

13. Applied egg-rr 95.8%

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

    1. unpow2 95.8%

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

    2. rem-sqrt-square 95.8%

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

    3. associate-/r* 95.8%

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

15. Simplified 95.8%

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

16. Final simplification 95.8%

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

Alternative 6: 95.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in f around 0 95.5%

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

   1. associate-/r* 95.5%

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

   2. distribute-rgt-out-- 95.5%

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

   3. metadata-eval 95.5%

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

   4. associate-/r* 95.5%

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

4. Simplified 95.5%

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

5. Taylor expanded in f around 0 96.1%

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

   1. associate-*r/ 96.1%

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

   2. +-commutative 96.1%

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

   3. metadata-eval 96.1%

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

   4. associate-*l/ 96.1%

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

   5. mul-1-neg 96.1%

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

   6. sub-neg 96.1%

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

   7. associate-*l/ 96.1%

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

   8. metadata-eval 96.1%

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

7. Simplified 96.1%

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

8. Final simplification 96.1%

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

Alternative 7: 95.7% accurate, 3.3× speedup

Math

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

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in f around inf 7.0%

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

   1. exp-prod 7.0%

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

   2. *-commutative 7.0%

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

   3. distribute-lft-neg-in 7.0%

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

   4. metadata-eval 7.0%

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

   5. *-commutative 7.0%

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

4. Simplified 7.0%

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

   1. associate-*l/ 7.0%

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

6. Applied egg-rr 96.6%

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

8. Simplified 96.6%

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

   1. add-exp-log 96.6%

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

10. Applied egg-rr 96.6%

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

11. Taylor expanded in f around 0 96.1%

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

    1. associate-*r/ 96.1%

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

    2. mul-1-neg 96.1%

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

    3. log-rec 96.1%

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

    4. log-rec 96.1%

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

    5. mul-1-neg 96.1%

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

    6. +-commutative 96.1%

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

    7. mul-1-neg 96.1%

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

    8. unsub-neg 96.1%

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

    9. metadata-eval 96.1%

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

    10. associate-*r/ 96.1%

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

    11. log-div 95.7%

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

    12. associate-*r/ 95.7%

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

13. Simplified 95.5%

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

14. Final simplification 95.5%

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

Alternative 8: 95.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in f around 0 95.5%

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

   1. associate-/r* 95.5%

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

   2. distribute-rgt-out-- 95.5%

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

   3. metadata-eval 95.5%

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

   4. associate-/r* 95.5%

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

4. Simplified 95.5%

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

5. Taylor expanded in f around 0 96.1%

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

7. Simplified 95.7%

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

8. Final simplification 95.7%

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

Alternative 9: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi} \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ (log 7.62939453125e-6) PI) (- 4.0)))

C

double code(double f) {
	return (log(7.62939453125e-6) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return (Math.log(7.62939453125e-6) / Math.PI) * -4.0;
}

Python

def code(f):
	return (math.log(7.62939453125e-6) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(log(7.62939453125e-6) / pi) * Float64(-4.0))
end

MATLAB

function tmp = code(f)
	tmp = (log(7.62939453125e-6) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[Log[7.62939453125e-6], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

3. Applied egg-rr 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{262144}}\right) \]

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}} \]

5. Final simplification 1.6%

   \[\leadsto \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi} \cdot \left(-4\right) \]

Reproduce

herbie shell --seed 2023217 
(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)))))))))