VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 96.4%

Time: 21.2s

Alternatives: 4

Speedup: 3.3×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

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: 96.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right)\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/ (log (fma f (* PI 0.08333333333333333) (/ 2.0 (* f (* PI 0.5))))) PI)))

C

double code(double f) {
	return -4.0 * (log(fma(f, (((double) M_PI) * 0.08333333333333333), (2.0 / (f * (((double) M_PI) * 0.5))))) / ((double) M_PI));
}

Julia

function code(f)
	return Float64(-4.0 * Float64(log(fma(f, Float64(pi * 0.08333333333333333), Float64(2.0 / Float64(f * Float64(pi * 0.5))))) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(2.0 / N[(f * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

   1. distribute-lft-neg-in 7.0%

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

   2. *-commutative 7.0%

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

3. Simplified 7.0%

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

4. Taylor expanded in f around inf 7.0%

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

5. Taylor expanded in f around 0 95.6%

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

7. Simplified 95.6%

   \[\leadsto -4 \cdot \frac{\log \color{blue}{\left(\mathsf{fma}\left(f, 0.125 \cdot \frac{{\pi}^{2}}{\pi} + \frac{{\pi}^{3}}{{\pi}^{2}} \cdot -0.041666666666666664, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right) + 0\right)}}{\pi} \]
8. Step-by-step derivation

   1. expm1-log1p-u 95.6%

      \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.125 \cdot \frac{{\pi}^{2}}{\pi} + \frac{{\pi}^{3}}{{\pi}^{2}} \cdot -0.041666666666666664\right)\right)}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right) + 0\right)}{\pi} \]

   2. expm1-udef 95.6%

      \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \color{blue}{e^{\mathsf{log1p}\left(0.125 \cdot \frac{{\pi}^{2}}{\pi} + \frac{{\pi}^{3}}{{\pi}^{2}} \cdot -0.041666666666666664\right)} - 1}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right) + 0\right)}{\pi} \]

9. Applied egg-rr 95.6%

   \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\pi, -0.041666666666666664, \pi \cdot 0.125\right)\right)} - 1}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right) + 0\right)}{\pi} \]
10. Step-by-step derivation

    1. expm1-def 95.6%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, -0.041666666666666664, \pi \cdot 0.125\right)\right)\right)}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right) + 0\right)}{\pi} \]

    2. expm1-log1p 95.6%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\mathsf{fma}\left(\pi, -0.041666666666666664, \pi \cdot 0.125\right)}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right) + 0\right)}{\pi} \]

    3. fma-udef 95.6%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot -0.041666666666666664 + \pi \cdot 0.125}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right) + 0\right)}{\pi} \]

    4. distribute-lft-out 95.6%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(-0.041666666666666664 + 0.125\right)}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right) + 0\right)}{\pi} \]

    5. metadata-eval 95.6%

       \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.08333333333333333}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right) + 0\right)}{\pi} \]

11. Simplified 95.6%

    \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot 0.08333333333333333}, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right) + 0\right)}{\pi} \]

12. Final simplification 95.6%

    \[\leadsto -4 \cdot \frac{\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right)\right)}{\pi} \]

Alternative 2: 96.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{4}{f \cdot \pi}\right)}{\pi} + -0.125 \cdot \left(\pi \cdot {f}^{2}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (+ (* -4.0 (/ (log (/ 4.0 (* f PI))) PI)) (* -0.125 (* PI (pow f 2.0)))))

C

double code(double f) {
	return (-4.0 * (log((4.0 / (f * ((double) M_PI)))) / ((double) M_PI))) + (-0.125 * (((double) M_PI) * pow(f, 2.0)));
}

Java

public static double code(double f) {
	return (-4.0 * (Math.log((4.0 / (f * Math.PI))) / Math.PI)) + (-0.125 * (Math.PI * Math.pow(f, 2.0)));
}

Python

def code(f):
	return (-4.0 * (math.log((4.0 / (f * math.pi))) / math.pi)) + (-0.125 * (math.pi * math.pow(f, 2.0)))

Julia

function code(f)
	return Float64(Float64(-4.0 * Float64(log(Float64(4.0 / Float64(f * pi))) / pi)) + Float64(-0.125 * Float64(pi * (f ^ 2.0))))
end

MATLAB

function tmp = code(f)
	tmp = (-4.0 * (log((4.0 / (f * pi))) / pi)) + (-0.125 * (pi * (f ^ 2.0)));
end

Wolfram

code[f_] := N[(N[(-4.0 * N[(N[Log[N[(4.0 / N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] + N[(-0.125 * N[(Pi * N[Power[f, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

   1. distribute-lft-neg-in 7.0%

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

   2. *-commutative 7.0%

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

3. Simplified 7.0%

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

4. Taylor expanded in f around 0 95.0%

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

   1. distribute-rgt-out-- 95.0%

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

   2. metadata-eval 95.0%

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

6. Simplified 95.0%

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

7. Taylor expanded in f around 0 95.2%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}{\pi} + -0.125 \cdot \left({f}^{2} \cdot \pi\right)} \]
8. Step-by-step derivation

   1. mul-1-neg 95.2%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(-\log f\right)}}{\pi} + -0.125 \cdot \left({f}^{2} \cdot \pi\right) \]

   2. neg-log 95.2%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + \color{blue}{\log \left(\frac{1}{f}\right)}}{\pi} + -0.125 \cdot \left({f}^{2} \cdot \pi\right) \]

   3. log-prod 95.2%

      \[\leadsto -4 \cdot \frac{\color{blue}{\log \left(\frac{4}{\pi} \cdot \frac{1}{f}\right)}}{\pi} + -0.125 \cdot \left({f}^{2} \cdot \pi\right) \]

   4. div-inv 95.2%

      \[\leadsto -4 \cdot \frac{\log \color{blue}{\left(\frac{\frac{4}{\pi}}{f}\right)}}{\pi} + -0.125 \cdot \left({f}^{2} \cdot \pi\right) \]

   5. associate-/l/ 95.2%

      \[\leadsto -4 \cdot \frac{\log \color{blue}{\left(\frac{4}{f \cdot \pi}\right)}}{\pi} + -0.125 \cdot \left({f}^{2} \cdot \pi\right) \]

   6. *-commutative 95.2%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{4}{\color{blue}{\pi \cdot f}}\right)}{\pi} + -0.125 \cdot \left({f}^{2} \cdot \pi\right) \]

9. Applied egg-rr 95.2%

   \[\leadsto -4 \cdot \frac{\color{blue}{\log \left(\frac{4}{\pi \cdot f}\right)}}{\pi} + -0.125 \cdot \left({f}^{2} \cdot \pi\right) \]

10. Final simplification 95.2%

    \[\leadsto -4 \cdot \frac{\log \left(\frac{4}{f \cdot \pi}\right)}{\pi} + -0.125 \cdot \left(\pi \cdot {f}^{2}\right) \]

Alternative 3: 95.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

   1. distribute-lft-neg-in 7.0%

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

   2. *-commutative 7.0%

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

3. Simplified 7.0%

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

4. Taylor expanded in f around 0 95.0%

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

   1. *-commutative 95.0%

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

   2. associate-/r* 95.0%

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

   3. distribute-rgt-out-- 95.0%

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

   4. metadata-eval 95.0%

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

   5. metadata-eval 95.0%

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

   6. associate-/r* 95.0%

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

   7. metadata-eval 95.0%

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

6. Simplified 95.0%

   \[\leadsto \log \color{blue}{\left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)} \cdot \frac{-4}{\pi} \]
7. Step-by-step derivation

   1. div-inv 95.0%

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

   2. div-inv 95.0%

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

   3. metadata-eval 95.0%

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

8. Applied egg-rr 95.0%

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

   1. associate-*r/ 95.0%

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

   2. *-rgt-identity 95.0%

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

   3. associate-*l/ 95.0%

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

   4. metadata-eval 95.0%

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

10. Simplified 95.0%

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

11. Final simplification 95.0%

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

Alternative 4: 95.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

   1. distribute-lft-neg-in 7.0%

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

   2. *-commutative 7.0%

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

3. Simplified 7.0%

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

4. Taylor expanded in f around 0 95.0%

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

   1. *-commutative 95.0%

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

   2. associate-/r* 95.0%

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

   3. distribute-rgt-out-- 95.0%

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

   4. metadata-eval 95.0%

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

   5. metadata-eval 95.0%

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

   6. associate-/r* 95.0%

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

   7. metadata-eval 95.0%

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

6. Simplified 95.0%

   \[\leadsto \log \color{blue}{\left(\frac{\frac{\frac{2}{\pi}}{0.5}}{f}\right)} \cdot \frac{-4}{\pi} \]
7. Step-by-step derivation

   1. div-inv 95.0%

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

   2. div-inv 95.0%

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

   3. metadata-eval 95.0%

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

8. Applied egg-rr 95.0%

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

   1. associate-*r/ 95.0%

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

   2. *-rgt-identity 95.0%

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

   3. associate-*l/ 95.0%

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

   4. metadata-eval 95.0%

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

10. Simplified 95.0%

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

    1. associate-*r/ 95.1%

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

    2. associate-/l/ 95.1%

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

    3. *-commutative 95.1%

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

12. Applied egg-rr 95.1%

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

13. Final simplification 95.1%

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

Reproduce

herbie shell --seed 2023331 
(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)))))))))