VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.0% → 96.2%

Time: 30.2s

Alternatives: 8

Speedup: 3.3×

• Unsound rule application detected in e-graph. Results may not be sound.

• could not determine a ground truth

  1. f = -3.32650474810479e+170

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t_0}\\ t_2 := e^{-t_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t_1 + t_2}{t_1 - t_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Initial Program: 7.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t_0}\\ t_2 := e^{-t_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t_1 + t_2}{t_1 - t_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Alternative 1: 96.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (- (log (fma f (* PI 0.08333333333333333) (/ 2.0 (* f (* PI 0.5))))))
  (* PI 0.25)))

C

double code(double f) {
	return -log(fma(f, (((double) M_PI) * 0.08333333333333333), (2.0 / (f * (((double) M_PI) * 0.5))))) / (((double) M_PI) * 0.25);
}

Julia

function code(f)
	return Float64(Float64(-log(fma(f, Float64(pi * 0.08333333333333333), Float64(2.0 / Float64(f * Float64(pi * 0.5)))))) / Float64(pi * 0.25))
end

Wolfram

code[f_] := N[((-N[Log[N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(2.0 / N[(f * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $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. Taylor expanded in f around 0 93.5%

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

   1. associate-+r+ 93.5%

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

   2. +-commutative 93.5%

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

4. Simplified 93.5%

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

   1. *-un-lft-identity 93.5%

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

   2. fma-udef 93.5%

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

   3. div-inv 93.5%

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

   4. metadata-eval 93.5%

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

   5. pow-div 93.5%

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

   6. metadata-eval 93.5%

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

6. Applied egg-rr 93.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{0.0625 \cdot \left(\pi \cdot 2\right) + \frac{-2}{{\pi}^{-1} \cdot 48}}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]
7. Step-by-step derivation

   1. +-commutative 93.5%

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

   2. metadata-eval 93.5%

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

   3. associate-*r/ 93.5%

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

   4. fma-def 93.5%

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

   5. associate-/r* 93.5%

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

   6. unpow-1 93.5%

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

   7. remove-double-div 93.5%

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

   8. *-commutative 93.5%

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

   9. associate-*l* 93.5%

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

   10. metadata-eval 93.5%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \frac{\pi}{48}, \pi \cdot \color{blue}{0.125}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

8. Simplified 93.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{fma}\left(-2, \frac{\pi}{48}, \pi \cdot 0.125\right)}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]
9. Step-by-step derivation

   1. associate-*l/ 93.6%

      \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \frac{\pi}{48}, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\frac{\pi}{4}}} \]

   2. *-un-lft-identity 93.6%

      \[\leadsto -\frac{\color{blue}{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \frac{\pi}{48}, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}}{\frac{\pi}{4}} \]

   3. div-inv 93.6%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \color{blue}{\pi \cdot \frac{1}{48}}, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\frac{\pi}{4}} \]

   4. metadata-eval 93.6%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot \color{blue}{0.020833333333333332}, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\frac{\pi}{4}} \]

   5. div-inv 93.6%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\color{blue}{\pi \cdot \frac{1}{4}}} \]

   6. metadata-eval 93.6%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot \color{blue}{0.25}} \]

10. Applied egg-rr 93.6%

    \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(-2, \pi \cdot 0.020833333333333332, \pi \cdot 0.125\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25}} \]
11. Step-by-step derivation

    1. fma-def 93.6%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{-2 \cdot \left(\pi \cdot 0.020833333333333332\right) + \pi \cdot 0.125}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25} \]

    2. *-commutative 93.6%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, -2 \cdot \color{blue}{\left(0.020833333333333332 \cdot \pi\right)} + \pi \cdot 0.125, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25} \]

    3. associate-*r* 93.6%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\left(-2 \cdot 0.020833333333333332\right) \cdot \pi} + \pi \cdot 0.125, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25} \]

    4. metadata-eval 93.6%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{-0.041666666666666664} \cdot \pi + \pi \cdot 0.125, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25} \]

    5. *-commutative 93.6%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, -0.041666666666666664 \cdot \pi + \color{blue}{0.125 \cdot \pi}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25} \]

    6. distribute-rgt-out 93.6%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(-0.041666666666666664 + 0.125\right)}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25} \]

    7. metadata-eval 93.6%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.08333333333333333}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25} \]

    8. *-commutative 93.6%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{2}{\color{blue}{\left(f \cdot 0.5\right) \cdot \pi}}\right)\right)}{\pi \cdot 0.25} \]

    9. associate-*l* 93.6%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{2}{\color{blue}{f \cdot \left(0.5 \cdot \pi\right)}}\right)\right)}{\pi \cdot 0.25} \]

12. Simplified 93.6%

    \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{2}{f \cdot \left(0.5 \cdot \pi\right)}\right)\right)}{\pi \cdot 0.25}} \]

13. Final simplification 93.6%

    \[\leadsto \frac{-\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right)\right)}{\pi \cdot 0.25} \]

Alternative 2: 95.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(\frac{4}{f \cdot \pi} + f \cdot \left(\pi \cdot 0.125\right)\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (- (log (+ (/ 4.0 (* f PI)) (* f (* PI 0.125))))) (* PI 0.25)))

C

double code(double f) {
	return -log(((4.0 / (f * ((double) M_PI))) + (f * (((double) M_PI) * 0.125)))) / (((double) M_PI) * 0.25);
}

Java

public static double code(double f) {
	return -Math.log(((4.0 / (f * Math.PI)) + (f * (Math.PI * 0.125)))) / (Math.PI * 0.25);
}

Python

def code(f):
	return -math.log(((4.0 / (f * math.pi)) + (f * (math.pi * 0.125)))) / (math.pi * 0.25)

Julia

function code(f)
	return Float64(Float64(-log(Float64(Float64(4.0 / Float64(f * pi)) + Float64(f * Float64(pi * 0.125))))) / Float64(pi * 0.25))
end

MATLAB

function tmp = code(f)
	tmp = -log(((4.0 / (f * pi)) + (f * (pi * 0.125)))) / (pi * 0.25);
end

Wolfram

code[f_] := N[((-N[Log[N[(N[(4.0 / N[(f * Pi), $MachinePrecision]), $MachinePrecision] + N[(f * N[(Pi * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $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. Taylor expanded in f around 0 93.2%

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

   1. distribute-rgt-out-- 93.2%

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

   2. metadata-eval 93.2%

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

4. Simplified 93.2%

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

5. Taylor expanded in f around 0 93.3%

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

   1. associate-*r/ 93.3%

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

   2. metadata-eval 93.3%

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

   3. associate-/r* 93.3%

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

   4. *-commutative 93.3%

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

7. Simplified 93.3%

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

   1. associate-*l/ 93.3%

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

   2. *-un-lft-identity 93.3%

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

   3. associate-/l/ 93.3%

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

   4. associate-*l* 93.3%

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

   5. div-inv 93.3%

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

   6. metadata-eval 93.3%

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

9. Applied egg-rr 93.3%

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

10. Final simplification 93.3%

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

Alternative 3: 95.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(4.0 * N[(N[(N[Log[f], $MachinePrecision] - N[Log[N[(2.0 / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $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. Taylor expanded in f around 0 93.2%

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

   1. mul-1-neg 93.2%

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

   2. unsub-neg 93.2%

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

   3. distribute-rgt-out-- 93.2%

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

   4. metadata-eval 93.2%

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

4. Simplified 93.2%

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

5. Final simplification 93.2%

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

Alternative 4: 95.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(N[(4.0 / Pi), $MachinePrecision] * N[(N[Log[f], $MachinePrecision] - N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision]), $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. Taylor expanded in f around 0 93.2%

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

   1. distribute-rgt-out-- 93.2%

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

   2. metadata-eval 93.2%

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

4. Simplified 93.2%

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

5. Taylor expanded in f around 0 93.3%

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

   1. associate-*r/ 93.3%

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

   2. metadata-eval 93.3%

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

   3. associate-/r* 93.3%

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

   4. *-commutative 93.3%

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

7. Simplified 93.3%

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

8. Taylor expanded in f around 0 93.2%

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

   1. fma-def 93.2%

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

   2. *-lft-identity 93.2%

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

   3. associate-*l/ 93.2%

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

   4. unpow-1 93.2%

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

   5. associate-*r* 93.2%

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

   6. unpow-1 93.2%

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

   7. associate-*r/ 93.2%

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

   8. metadata-eval 93.2%

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

   9. fma-def 93.2%

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

   10. neg-mul-1 93.2%

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

   11. +-commutative 93.2%

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

   12. unsub-neg 93.2%

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

   13. log-div 93.1%

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

   14. associate-/r* 93.1%

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

   15. associate-/l/ 93.1%

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

10. Simplified 93.1%

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

11. Taylor expanded in f around 0 93.2%

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

    1. neg-mul-1 93.2%

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

    2. log-rec 93.2%

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

    3. +-commutative 93.2%

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

    4. log-rec 93.2%

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

    5. sub-neg 93.2%

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

13. Simplified 93.2%

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

14. Final simplification 93.2%

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

Alternative 5: 1.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(N[(N[Log[N[(f * N[(Pi * 0.125), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $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. Taylor expanded in f around 0 93.2%

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

   1. distribute-rgt-out-- 93.2%

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

   2. metadata-eval 93.2%

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

4. Simplified 93.2%

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

5. Taylor expanded in f around 0 93.3%

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

   1. associate-*r/ 93.3%

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

   2. metadata-eval 93.3%

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

   3. associate-/r* 93.3%

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

   4. *-commutative 93.3%

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

7. Simplified 93.3%

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

   1. associate-*l/ 93.3%

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

   2. *-un-lft-identity 93.3%

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

   3. associate-/l/ 93.3%

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

   4. associate-*l* 93.3%

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

   5. div-inv 93.3%

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

   6. metadata-eval 93.3%

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

9. Applied egg-rr 93.3%

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

10. Taylor expanded in f around inf 1.8%

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

    1. +-commutative 1.8%

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

    2. mul-1-neg 1.8%

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

    3. log-rec 1.8%

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

    4. remove-double-neg 1.8%

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

    5. *-commutative 1.8%

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

    6. log-prod 1.8%

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

12. Simplified 1.8%

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

13. Final simplification 1.8%

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

Alternative 6: 95.4% 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 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. Taylor expanded in f around 0 93.2%

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

   1. distribute-rgt-out-- 93.2%

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

   2. metadata-eval 93.2%

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

4. Simplified 93.2%

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

5. Taylor expanded in f around 0 93.3%

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

   1. associate-*r/ 93.3%

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

   2. metadata-eval 93.3%

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

   3. associate-/r* 93.3%

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

   4. *-commutative 93.3%

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

7. Simplified 93.3%

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

8. Taylor expanded in f around 0 93.2%

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

   1. fma-def 93.2%

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

   2. *-lft-identity 93.2%

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

   3. associate-*l/ 93.2%

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

   4. unpow-1 93.2%

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

   5. associate-*r* 93.2%

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

   6. unpow-1 93.2%

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

   7. associate-*r/ 93.2%

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

   8. metadata-eval 93.2%

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

   9. fma-def 93.2%

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

   10. neg-mul-1 93.2%

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

   11. +-commutative 93.2%

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

   12. unsub-neg 93.2%

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

   13. log-div 93.1%

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

   14. associate-/r* 93.1%

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

   15. associate-/l/ 93.1%

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

10. Simplified 93.1%

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

11. Final simplification 93.1%

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

Alternative 7: 95.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[((-N[Log[N[(4.0 / N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $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. Taylor expanded in f around 0 93.1%

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

   1. distribute-rgt-out-- 93.1%

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

   2. metadata-eval 93.1%

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

4. Simplified 93.1%

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

5. Taylor expanded in f around 0 93.2%

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

   1. metadata-eval 93.2%

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

   2. mul-1-neg 93.2%

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

   3. log-rec 93.2%

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

7. Simplified 93.1%

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

8. Final simplification 93.1%

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

Alternative 8: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{-\log 0.07407407407407407}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* 4.0 (/ (- (log 0.07407407407407407)) PI)))

C

double code(double f) {
	return 4.0 * (-log(0.07407407407407407) / ((double) M_PI));
}

Java

public static double code(double f) {
	return 4.0 * (-Math.log(0.07407407407407407) / Math.PI);
}

Python

def code(f):
	return 4.0 * (-math.log(0.07407407407407407) / math.pi)

Julia

function code(f)
	return Float64(4.0 * Float64(Float64(-log(0.07407407407407407)) / pi))
end

MATLAB

function tmp = code(f)
	tmp = 4.0 * (-log(0.07407407407407407) / pi);
end

Wolfram

code[f_] := N[(4.0 * N[((-N[Log[0.07407407407407407], $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 1.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{27}}\right) \]

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.07407407407407407}{\pi}} \]

5. Final simplification 1.6%

   \[\leadsto 4 \cdot \frac{-\log 0.07407407407407407}{\pi} \]

Reproduce

herbie shell --seed 2023178 
(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)))))))))