VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 96.6%

Time: 31.7s

Alternatives: 7

Speedup: 3.3×

• could not determine a ground truth

  1. f = -1.875893698562359e+158

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: 96.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ -\frac{\log \left(\frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\frac{\mathsf{fma}\left({\pi}^{5}, {f}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \pi \cdot \left(f \cdot 0.5\right) + 0.005208333333333333 \cdot {\left(\pi \cdot f\right)}^{3}\right)}{2}}\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (/
   (log
    (/
     (cosh (* PI (* 0.25 f)))
     (/
      (fma
       (pow PI 5.0)
       (* (pow f 5.0) 1.6276041666666666e-5)
       (+ (* PI (* f 0.5)) (* 0.005208333333333333 (pow (* PI f) 3.0))))
      2.0)))
   (* PI 0.25))))

C

double code(double f) {
	return -(log((cosh((((double) M_PI) * (0.25 * f))) / (fma(pow(((double) M_PI), 5.0), (pow(f, 5.0) * 1.6276041666666666e-5), ((((double) M_PI) * (f * 0.5)) + (0.005208333333333333 * pow((((double) M_PI) * f), 3.0)))) / 2.0))) / (((double) M_PI) * 0.25));
}

Julia

function code(f)
	return Float64(-Float64(log(Float64(cosh(Float64(pi * Float64(0.25 * f))) / Float64(fma((pi ^ 5.0), Float64((f ^ 5.0) * 1.6276041666666666e-5), Float64(Float64(pi * Float64(f * 0.5)) + Float64(0.005208333333333333 * (Float64(pi * f) ^ 3.0)))) / 2.0))) / Float64(pi * 0.25)))
end

Wolfram

code[f_] := (-N[(N[Log[N[(N[Cosh[N[(Pi * N[(0.25 * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[Power[Pi, 5.0], $MachinePrecision] * N[(N[Power[f, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision] + N[(N[(Pi * N[(f * 0.5), $MachinePrecision]), $MachinePrecision] + N[(0.005208333333333333 * N[Power[N[(Pi * f), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 6.5%

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

2. Taylor expanded in f around 0 97.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}}\right) \]

4. Simplified 97.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)}}\right) \]
5. Step-by-step derivation

   1. associate-*l/ 98.1%

      \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)}\right)}{\frac{\pi}{4}}} \]

6. Applied egg-rr 98.1%

   \[\leadsto -\color{blue}{\frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)}\right)}{\pi \cdot 0.25}} \]

8. Simplified 98.1%

   \[\leadsto -\color{blue}{\frac{\log \left(\frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\frac{\mathsf{fma}\left({\pi}^{5}, {f}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \pi \cdot 0.5, {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333\right)\right)}{2}}\right)}{\pi \cdot 0.25}} \]
9. Step-by-step derivation

   1. fma-udef 98.1%

      \[\leadsto -\frac{\log \left(\frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\frac{\mathsf{fma}\left({\pi}^{5}, {f}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{f \cdot \left(\pi \cdot 0.5\right) + {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333}\right)}{2}}\right)}{\pi \cdot 0.25} \]

   2. *-commutative 98.1%

      \[\leadsto -\frac{\log \left(\frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\frac{\mathsf{fma}\left({\pi}^{5}, {f}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{\left(\pi \cdot 0.5\right) \cdot f} + {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333\right)}{2}}\right)}{\pi \cdot 0.25} \]

   3. associate-*l* 98.1%

      \[\leadsto -\frac{\log \left(\frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\frac{\mathsf{fma}\left({\pi}^{5}, {f}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{\pi \cdot \left(0.5 \cdot f\right)} + {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333\right)}{2}}\right)}{\pi \cdot 0.25} \]

   4. *-commutative 98.1%

      \[\leadsto -\frac{\log \left(\frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\frac{\mathsf{fma}\left({\pi}^{5}, {f}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \pi \cdot \left(0.5 \cdot f\right) + {\color{blue}{\left(\pi \cdot f\right)}}^{3} \cdot 0.005208333333333333\right)}{2}}\right)}{\pi \cdot 0.25} \]

10. Applied egg-rr 98.1%

    \[\leadsto -\frac{\log \left(\frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\frac{\mathsf{fma}\left({\pi}^{5}, {f}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{\pi \cdot \left(0.5 \cdot f\right) + {\left(\pi \cdot f\right)}^{3} \cdot 0.005208333333333333}\right)}{2}}\right)}{\pi \cdot 0.25} \]

11. Final simplification 98.1%

    \[\leadsto -\frac{\log \left(\frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\frac{\mathsf{fma}\left({\pi}^{5}, {f}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \pi \cdot \left(f \cdot 0.5\right) + 0.005208333333333333 \cdot {\left(\pi \cdot f\right)}^{3}\right)}{2}}\right)}{\pi \cdot 0.25} \]

Alternative 2: 96.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(-\frac{\log \left(\left(\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right) \cdot 2\right) \cdot \frac{1}{\mathsf{fma}\left(f, \pi \cdot 0.5, 0.005208333333333333 \cdot {\left(\pi \cdot f\right)}^{3}\right)}\right)}{\pi}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  4.0
  (-
   (/
    (log
     (*
      (* (cosh (* PI (* 0.25 f))) 2.0)
      (/ 1.0 (fma f (* PI 0.5) (* 0.005208333333333333 (pow (* PI f) 3.0))))))
    PI))))

C

double code(double f) {
	return 4.0 * -(log(((cosh((((double) M_PI) * (0.25 * f))) * 2.0) * (1.0 / fma(f, (((double) M_PI) * 0.5), (0.005208333333333333 * pow((((double) M_PI) * f), 3.0)))))) / ((double) M_PI));
}

Julia

function code(f)
	return Float64(4.0 * Float64(-Float64(log(Float64(Float64(cosh(Float64(pi * Float64(0.25 * f))) * 2.0) * Float64(1.0 / fma(f, Float64(pi * 0.5), Float64(0.005208333333333333 * (Float64(pi * f) ^ 3.0)))))) / pi)))
end

Wolfram

code[f_] := N[(4.0 * (-N[(N[Log[N[(N[(N[Cosh[N[(Pi * N[(0.25 * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * N[(1.0 / N[(f * N[(Pi * 0.5), $MachinePrecision] + N[(0.005208333333333333 * N[Power[N[(Pi * f), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

2. Taylor expanded in f around inf 6.5%

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

   1. exp-prod 6.5%

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

   2. *-commutative 6.5%

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

   3. distribute-lft-neg-in 6.5%

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

   4. metadata-eval 6.5%

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

   5. *-commutative 6.5%

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

4. Simplified 6.5%

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

5. Taylor expanded in f around inf 6.5%

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

   1. div-inv 6.5%

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

   2. cosh-undef 6.5%

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

   3. associate-*r* 6.5%

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

   4. *-commutative 6.5%

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

   5. exp-prod 6.5%

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

   6. *-commutative 6.5%

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

   7. exp-prod 6.5%

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

   8. *-commutative 6.5%

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

7. Applied egg-rr 6.5%

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

8. Taylor expanded in f around 0 97.9%

   \[\leadsto -4 \cdot \frac{\log \left(\left(2 \cdot \cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)\right) \cdot \frac{1}{\color{blue}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}}\right)}{\pi} \]
9. Step-by-step derivation

   1. *-commutative 97.9%

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

   2. fma-def 97.9%

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

   3. distribute-rgt-out-- 97.9%

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

   4. metadata-eval 97.9%

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

   5. distribute-rgt-out-- 97.9%

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

   6. metadata-eval 97.9%

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

   7. associate-*r* 97.9%

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

   8. cube-prod 97.9%

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

10. Simplified 97.9%

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

11. Final simplification 97.9%

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

Alternative 3: 96.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

2. Taylor expanded in f around 0 97.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}}\right) \]

4. Simplified 97.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)}}\right) \]
5. Step-by-step derivation

   1. associate-*l/ 98.1%

      \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)}\right)}{\frac{\pi}{4}}} \]

6. Applied egg-rr 98.1%

   \[\leadsto -\color{blue}{\frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)}\right)}{\pi \cdot 0.25}} \]

8. Simplified 98.1%

   \[\leadsto -\color{blue}{\frac{\log \left(\frac{\cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\frac{\mathsf{fma}\left({\pi}^{5}, {f}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(f, \pi \cdot 0.5, {\left(f \cdot \pi\right)}^{3} \cdot 0.005208333333333333\right)\right)}{2}}\right)}{\pi \cdot 0.25}} \]

9. Taylor expanded in f around 0 97.8%

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

    1. +-commutative 97.8%

       \[\leadsto -\color{blue}{\left(4 \cdot \frac{-1 \cdot \log f + \log \left(\frac{4}{\pi}\right)}{\pi} + 2 \cdot \left({f}^{2} \cdot \left(0.0625 \cdot \pi - 0.020833333333333332 \cdot \pi\right)\right)\right)} \]

    2. fma-def 97.8%

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

    3. +-commutative 97.8%

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

    4. mul-1-neg 97.8%

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

    5. unsub-neg 97.8%

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

    6. log-div 97.8%

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

    7. associate-/r* 97.8%

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

    8. *-commutative 97.8%

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

    9. *-commutative 97.8%

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

    10. distribute-rgt-out-- 97.8%

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

11. Simplified 97.8%

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

12. Final simplification 97.8%

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

Alternative 4: 95.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (- (fma 4.0 (/ (log (/ 4.0 (* PI f))) PI) (* PI (* (* f f) 0.0625)))))

C

double code(double f) {
	return -fma(4.0, (log((4.0 / (((double) M_PI) * f))) / ((double) M_PI)), (((double) M_PI) * ((f * f) * 0.0625)));
}

Julia

function code(f)
	return Float64(-fma(4.0, Float64(log(Float64(4.0 / Float64(pi * f))) / pi), Float64(pi * Float64(Float64(f * f) * 0.0625))))
end

Wolfram

code[f_] := (-N[(4.0 * N[(N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] + N[(Pi * N[(N[(f * f), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 6.5%

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

2. Taylor expanded in f around 0 97.2%

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

   1. distribute-rgt-out-- 97.2%

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

   2. metadata-eval 97.2%

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

4. Simplified 97.2%

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

5. Taylor expanded in f around 0 97.3%

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

6. Taylor expanded in f around 0 97.4%

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

   1. fma-def 97.4%

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

   2. +-commutative 97.4%

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

   3. mul-1-neg 97.4%

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

   4. unsub-neg 97.4%

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

   5. log-div 97.4%

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

   6. associate-/r* 97.4%

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

   7. *-commutative 97.4%

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

   8. *-commutative 97.4%

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

   9. *-commutative 97.4%

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

   10. associate-*l* 97.4%

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

   11. unpow2 97.4%

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

8. Simplified 97.4%

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

9. Final simplification 97.4%

   \[\leadsto -\mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi}, \pi \cdot \left(\left(f \cdot f\right) \cdot 0.0625\right)\right) \]

Alternative 5: 95.8% accurate, 2.5× speedup

Math

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

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

Derivation

1. Initial program 6.5%

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

2. Taylor expanded in f around 0 97.2%

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

   1. mul-1-neg 97.2%

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

   2. unsub-neg 97.2%

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

   3. distribute-rgt-out-- 97.2%

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

   4. metadata-eval 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-/r* 97.2%

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

   7. metadata-eval 97.2%

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

4. Simplified 97.2%

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

5. Final simplification 97.2%

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

Alternative 6: 95.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

2. Taylor expanded in f around 0 97.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}}\right) \]

4. Simplified 97.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)}}\right) \]

5. Taylor expanded in f around 0 97.2%

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

   1. associate-*r/ 97.2%

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

   2. neg-mul-1 97.2%

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

   3. log-rec 97.2%

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

   4. +-commutative 97.2%

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

   5. log-rec 97.2%

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

   6. unsub-neg 97.2%

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

   7. log-div 97.2%

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

   8. associate-*r/ 97.2%

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

7. Simplified 97.2%

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

8. Final simplification 97.2%

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

Alternative 7: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{-4}{\frac{\pi}{\log 0.03125}} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (- 4.0) (/ PI (log 0.03125))))

C

double code(double f) {
	return -4.0 / (((double) M_PI) / log(0.03125));
}

Java

public static double code(double f) {
	return -4.0 / (Math.PI / Math.log(0.03125));
}

Python

def code(f):
	return -4.0 / (math.pi / math.log(0.03125))

Julia

function code(f)
	return Float64(Float64(-4.0) / Float64(pi / log(0.03125)))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 / (pi / log(0.03125));
end

Wolfram

code[f_] := N[((-4.0) / N[(Pi / N[Log[0.03125], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

3. Applied egg-rr 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{64}}\right) \]

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.03125}{\pi}} \]
5. Step-by-step derivation

   1. associate-*r/ 1.6%

      \[\leadsto -\color{blue}{\frac{4 \cdot \log 0.03125}{\pi}} \]

   2. associate-/l* 1.6%

      \[\leadsto -\color{blue}{\frac{4}{\frac{\pi}{\log 0.03125}}} \]

6. Simplified 1.6%

   \[\leadsto -\color{blue}{\frac{4}{\frac{\pi}{\log 0.03125}}} \]

7. Final simplification 1.6%

   \[\leadsto \frac{-4}{\frac{\pi}{\log 0.03125}} \]

Reproduce

herbie shell --seed 2023228 
(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)))))))))