VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 96.6%

Time: 30.3s

Alternatives: 5

Speedup: 3.3×

• could not determine a ground truth

  1. f = -1.3635196891535975e+54

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: 6.8% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(\log \left({\left(e^{-0.25}\right)}^{\left(f \cdot \pi\right)} + {\left(e^{0.25}\right)}^{\left(f \cdot \pi\right)}\right) - \log \left(\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (-
   (log (+ (pow (exp -0.25) (* f PI)) (pow (exp 0.25) (* f PI))))
   (log
    (fma
     f
     (* PI 0.5)
     (fma
      (pow f 3.0)
      (* (pow PI 3.0) 0.005208333333333333)
      (* (pow f 5.0) (* (pow PI 5.0) 1.6276041666666666e-5))))))
  (/ -4.0 PI)))

C

double code(double f) {
	return (log((pow(exp(-0.25), (f * ((double) M_PI))) + pow(exp(0.25), (f * ((double) M_PI))))) - log(fma(f, (((double) M_PI) * 0.5), fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), (pow(f, 5.0) * (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5)))))) * (-4.0 / ((double) M_PI));
}

Julia

function code(f)
	return Float64(Float64(log(Float64((exp(-0.25) ^ Float64(f * pi)) + (exp(0.25) ^ Float64(f * pi)))) - log(fma(f, Float64(pi * 0.5), fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), Float64((f ^ 5.0) * Float64((pi ^ 5.0) * 1.6276041666666666e-5)))))) * Float64(-4.0 / pi))
end

Wolfram

code[f_] := N[(N[(N[Log[N[(N[Power[N[Exp[-0.25], $MachinePrecision], N[(f * Pi), $MachinePrecision]], $MachinePrecision] + N[Power[N[Exp[0.25], $MachinePrecision], N[(f * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[N[(f * N[(Pi * 0.5), $MachinePrecision] + N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.1%

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

   1. *-commutative 6.1%

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

   2. distribute-rgt-neg-in 6.1%

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

   3. associate-/r/ 6.1%

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

   4. associate-*l/ 6.1%

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

   5. metadata-eval 6.1%

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

   6. distribute-neg-frac 6.1%

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

3. Simplified 6.1%

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

4. Taylor expanded in f around inf 6.1%

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

5. Taylor expanded in f around 0 95.5%

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

   1. log-div 95.8%

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

   2. exp-prod 95.8%

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

   3. exp-prod 95.8%

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

   4. fma-def 95.8%

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

7. Applied egg-rr 95.8%

   \[\leadsto \color{blue}{\left(\log \left({\left(e^{-0.25}\right)}^{\left(f \cdot \pi\right)} + {\left(e^{0.25}\right)}^{\left(f \cdot \pi\right)}\right) - \log \left(\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]

8. Final simplification 95.8%

   \[\leadsto \left(\log \left({\left(e^{-0.25}\right)}^{\left(f \cdot \pi\right)} + {\left(e^{0.25}\right)}^{\left(f \cdot \pi\right)}\right) - \log \left(\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)\right)\right)\right) \cdot \frac{-4}{\pi} \]

Alternative 2: 96.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{f \cdot f}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), 0\right), -2, 0\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (fma
  (/ (- (log (/ (/ 2.0 PI) 0.5)) (log f)) PI)
  -4.0
  (fma
   (*
    (/ (* f f) PI)
    (fma
     PI
     (* 0.5 (fma (* PI 0.020833333333333332) -2.0 (* 0.0625 (/ PI 0.5))))
     0.0))
   -2.0
   0.0)))

C

double code(double f) {
	return fma(((log(((2.0 / ((double) M_PI)) / 0.5)) - log(f)) / ((double) M_PI)), -4.0, fma((((f * f) / ((double) M_PI)) * fma(((double) M_PI), (0.5 * fma((((double) M_PI) * 0.020833333333333332), -2.0, (0.0625 * (((double) M_PI) / 0.5)))), 0.0)), -2.0, 0.0));
}

Julia

function code(f)
	return fma(Float64(Float64(log(Float64(Float64(2.0 / pi) / 0.5)) - log(f)) / pi), -4.0, fma(Float64(Float64(Float64(f * f) / pi) * fma(pi, Float64(0.5 * fma(Float64(pi * 0.020833333333333332), -2.0, Float64(0.0625 * Float64(pi / 0.5)))), 0.0)), -2.0, 0.0))
end

Wolfram

code[f_] := N[(N[(N[(N[Log[N[(N[(2.0 / Pi), $MachinePrecision] / 0.5), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * -4.0 + N[(N[(N[(N[(f * f), $MachinePrecision] / Pi), $MachinePrecision] * N[(Pi * N[(0.5 * N[(N[(Pi * 0.020833333333333332), $MachinePrecision] * -2.0 + N[(0.0625 * N[(Pi / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision] * -2.0 + 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.1%

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

   1. *-commutative 6.1%

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

   2. distribute-rgt-neg-in 6.1%

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

   3. associate-/r/ 6.1%

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

   4. associate-*l/ 6.1%

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

   5. metadata-eval 6.1%

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

   6. distribute-neg-frac 6.1%

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

3. Simplified 6.1%

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

4. Taylor expanded in f around 0 95.7%

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

6. Simplified 95.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{f \cdot f}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, 0.0625 \cdot \left(\frac{\pi}{0.5} \cdot 1\right)\right), 0\right), -2, 0\right)\right)} \]
7. Step-by-step derivation

   1. pow1 95.7%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{f \cdot f}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\color{blue}{{\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right)}^{1}}, -2, 0.0625 \cdot \left(\frac{\pi}{0.5} \cdot 1\right)\right), 0\right), -2, 0\right)\right) \]

   2. pow-div 95.7%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{f \cdot f}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left({\left(\color{blue}{{\pi}^{\left(3 - 2\right)}} \cdot 0.020833333333333332\right)}^{1}, -2, 0.0625 \cdot \left(\frac{\pi}{0.5} \cdot 1\right)\right), 0\right), -2, 0\right)\right) \]

   3. metadata-eval 95.7%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{f \cdot f}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left({\left({\pi}^{\color{blue}{1}} \cdot 0.020833333333333332\right)}^{1}, -2, 0.0625 \cdot \left(\frac{\pi}{0.5} \cdot 1\right)\right), 0\right), -2, 0\right)\right) \]

   4. pow1 95.7%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{f \cdot f}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left({\left(\color{blue}{\pi} \cdot 0.020833333333333332\right)}^{1}, -2, 0.0625 \cdot \left(\frac{\pi}{0.5} \cdot 1\right)\right), 0\right), -2, 0\right)\right) \]

8. Applied egg-rr 95.7%

   \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{f \cdot f}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\color{blue}{{\left(\pi \cdot 0.020833333333333332\right)}^{1}}, -2, 0.0625 \cdot \left(\frac{\pi}{0.5} \cdot 1\right)\right), 0\right), -2, 0\right)\right) \]
9. Step-by-step derivation

   1. unpow1 95.7%

      \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{f \cdot f}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\color{blue}{\pi \cdot 0.020833333333333332}, -2, 0.0625 \cdot \left(\frac{\pi}{0.5} \cdot 1\right)\right), 0\right), -2, 0\right)\right) \]

10. Simplified 95.7%

    \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{f \cdot f}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\color{blue}{\pi \cdot 0.020833333333333332}, -2, 0.0625 \cdot \left(\frac{\pi}{0.5} \cdot 1\right)\right), 0\right), -2, 0\right)\right) \]

11. Final simplification 95.7%

    \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{\frac{2}{\pi}}{0.5}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{f \cdot f}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, 0.0625 \cdot \frac{\pi}{0.5}\right), 0\right), -2, 0\right)\right) \]

Alternative 3: 96.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.1%

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

   1. *-commutative 6.1%

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

   2. distribute-rgt-neg-in 6.1%

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

   3. associate-/r/ 6.1%

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

   4. associate-*l/ 6.1%

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

   5. metadata-eval 6.1%

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

   6. distribute-neg-frac 6.1%

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

3. Simplified 6.1%

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

4. Taylor expanded in f around inf 6.1%

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

5. Taylor expanded in f around 0 95.3%

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

   1. distribute-rgt-out-- 95.3%

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

   2. metadata-eval 95.3%

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

   3. neg-mul-1 95.3%

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

   4. unsub-neg 95.3%

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

   5. log-div 95.0%

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

   6. *-commutative 95.0%

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

   7. associate-/r* 95.0%

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

   8. metadata-eval 95.0%

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

7. Simplified 95.0%

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

8. Taylor expanded in f around 0 95.3%

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

   1. mul-1-neg 95.3%

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

   2. sub-neg 95.3%

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

10. Simplified 95.3%

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

11. Final simplification 95.3%

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

Alternative 4: 96.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.1%

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

   1. *-commutative 6.1%

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

   2. distribute-rgt-neg-in 6.1%

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

   3. associate-/r/ 6.1%

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

   4. associate-*l/ 6.1%

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

   5. metadata-eval 6.1%

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

   6. distribute-neg-frac 6.1%

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

3. Simplified 6.1%

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

4. Taylor expanded in f around inf 6.1%

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

5. Taylor expanded in f around 0 95.3%

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

   1. distribute-rgt-out-- 95.3%

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

   2. metadata-eval 95.3%

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

   3. neg-mul-1 95.3%

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

   4. unsub-neg 95.3%

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

   5. log-div 95.0%

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

   6. *-commutative 95.0%

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

   7. associate-/r* 95.0%

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

   8. metadata-eval 95.0%

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

7. Simplified 95.0%

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

8. Final simplification 95.0%

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

Alternative 5: 1.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.1%

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

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%

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

Reproduce

herbie shell --seed 2023272 
(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)))))))))