VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.0%

Time: 23.2s

Alternatives: 8

Speedup: 4.9×

• 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: 6.7% 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: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log (- (/ 1.0 (expm1 (* 0.5 (* PI f)))) (/ 1.0 (expm1 (* PI (* f -0.5))))))
   PI)))

C

double code(double f) {
	return -4.0 * (log(((1.0 / expm1((0.5 * (((double) M_PI) * f)))) - (1.0 / expm1((((double) M_PI) * (f * -0.5)))))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log(((1.0 / Math.expm1((0.5 * (Math.PI * f)))) - (1.0 / Math.expm1((Math.PI * (f * -0.5)))))) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log(((1.0 / math.expm1((0.5 * (math.pi * f)))) - (1.0 / math.expm1((math.pi * (f * -0.5)))))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log(Float64(Float64(1.0 / expm1(Float64(0.5 * Float64(pi * f)))) - Float64(1.0 / expm1(Float64(pi * Float64(f * -0.5)))))) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(N[(1.0 / N[(Exp[N[(0.5 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(Exp[N[(Pi * N[(f * -0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.4%

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

3. Simplified 98.0%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around inf 6.1%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{e^{0.5 \cdot \left(f \cdot \pi\right)} - 1} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi}} \]
6. Step-by-step derivation

   1. expm1-define 6.3%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\color{blue}{\mathsf{expm1}\left(0.5 \cdot \left(f \cdot \pi\right)\right)}} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   2. *-commutative 6.3%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \color{blue}{\left(\pi \cdot f\right)}\right)} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   3. expm1-define 98.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} - \frac{1}{\color{blue}{\mathsf{expm1}\left(-0.5 \cdot \left(f \cdot \pi\right)\right)}}\right)}{\pi} \]

   4. associate-*r* 98.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\left(-0.5 \cdot f\right) \cdot \pi}\right)}\right)}{\pi} \]

   5. *-commutative 98.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\left(f \cdot -0.5\right)} \cdot \pi\right)}\right)}{\pi} \]

   6. *-commutative 98.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\pi \cdot \left(f \cdot -0.5\right)}\right)}\right)}{\pi} \]

7. Simplified 98.1%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}} \]
8. Add Preprocessing

Alternative 2: 96.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} + -0.08333333333333333 \cdot \left(\pi \cdot {f}^{2}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (+
  (* -4.0 (/ (- (log (/ 4.0 PI)) (log f)) PI))
  (* -0.08333333333333333 (* PI (pow f 2.0)))))

C

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

Java

public static double code(double f) {
	return (-4.0 * ((Math.log((4.0 / Math.PI)) - Math.log(f)) / Math.PI)) + (-0.08333333333333333 * (Math.PI * Math.pow(f, 2.0)));
}

Python

def code(f):
	return (-4.0 * ((math.log((4.0 / math.pi)) - math.log(f)) / math.pi)) + (-0.08333333333333333 * (math.pi * math.pow(f, 2.0)))

Julia

function code(f)
	return Float64(Float64(-4.0 * Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi)) + Float64(-0.08333333333333333 * Float64(pi * (f ^ 2.0))))
end

MATLAB

function tmp = code(f)
	tmp = (-4.0 * ((log((4.0 / pi)) - log(f)) / pi)) + (-0.08333333333333333 * (pi * (f ^ 2.0)));
end

Wolfram

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

Derivation

1. Initial program 5.4%

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

3. Simplified 98.0%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 94.4%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + 0.25 \cdot \left({f}^{2} \cdot \left(\pi \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]
6. Step-by-step derivation

   1. +-commutative 94.4%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(0.25 \cdot \left({f}^{2} \cdot \left(\pi \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)\right) + -1 \cdot \log f\right)}\right) \cdot \frac{-4}{\pi} \]

   2. mul-1-neg 94.4%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \left(0.25 \cdot \left({f}^{2} \cdot \left(\pi \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)\right) + \color{blue}{\left(-\log f\right)}\right)\right) \cdot \frac{-4}{\pi} \]

7. Simplified 94.4%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(0.25 \cdot \left({f}^{2} \cdot \left(0.08333333333333333 \cdot {\pi}^{2}\right)\right) - \log f\right)\right)} \cdot \frac{-4}{\pi} \]

8. Taylor expanded in f around 0 94.5%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} + -0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)} \]

9. Final simplification 94.5%

   \[\leadsto -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} + -0.08333333333333333 \cdot \left(\pi \cdot {f}^{2}\right) \]
10. Add Preprocessing

Alternative 3: 96.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ -0.08333333333333333 \cdot \left(\pi \cdot {f}^{2}\right) + -4 \cdot \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (+
  (* -0.08333333333333333 (* PI (pow f 2.0)))
  (* -4.0 (/ (log (/ (/ 4.0 PI) f)) PI))))

C

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

Java

public static double code(double f) {
	return (-0.08333333333333333 * (Math.PI * Math.pow(f, 2.0))) + (-4.0 * (Math.log(((4.0 / Math.PI) / f)) / Math.PI));
}

Python

def code(f):
	return (-0.08333333333333333 * (math.pi * math.pow(f, 2.0))) + (-4.0 * (math.log(((4.0 / math.pi) / f)) / math.pi))

Julia

function code(f)
	return Float64(Float64(-0.08333333333333333 * Float64(pi * (f ^ 2.0))) + Float64(-4.0 * Float64(log(Float64(Float64(4.0 / pi) / f)) / pi)))
end

MATLAB

function tmp = code(f)
	tmp = (-0.08333333333333333 * (pi * (f ^ 2.0))) + (-4.0 * (log(((4.0 / pi) / f)) / pi));
end

Wolfram

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

Derivation

1. Initial program 5.4%

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

3. Simplified 98.0%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 94.4%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + 0.25 \cdot \left({f}^{2} \cdot \left(\pi \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]
6. Step-by-step derivation

   1. +-commutative 94.4%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(0.25 \cdot \left({f}^{2} \cdot \left(\pi \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)\right) + -1 \cdot \log f\right)}\right) \cdot \frac{-4}{\pi} \]

   2. mul-1-neg 94.4%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \left(0.25 \cdot \left({f}^{2} \cdot \left(\pi \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)\right) + \color{blue}{\left(-\log f\right)}\right)\right) \cdot \frac{-4}{\pi} \]

7. Simplified 94.4%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(0.25 \cdot \left({f}^{2} \cdot \left(0.08333333333333333 \cdot {\pi}^{2}\right)\right) - \log f\right)\right)} \cdot \frac{-4}{\pi} \]

8. Taylor expanded in f around 0 94.5%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} + -0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)} \]
9. Step-by-step derivation

   1. diff-log 94.4%

      \[\leadsto -4 \cdot \frac{\color{blue}{\log \left(\frac{\frac{4}{\pi}}{f}\right)}}{\pi} + -0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right) \]

10. Applied egg-rr 94.4%

    \[\leadsto -4 \cdot \frac{\color{blue}{\log \left(\frac{\frac{4}{\pi}}{f}\right)}}{\pi} + -0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right) \]

11. Final simplification 94.4%

    \[\leadsto -0.08333333333333333 \cdot \left(\pi \cdot {f}^{2}\right) + -4 \cdot \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi} \]
12. Add Preprocessing

Alternative 4: 96.0% 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 5.4%

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

3. Simplified 98.0%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 94.2%

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

   1. mul-1-neg 94.2%

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

   2. unsub-neg 94.2%

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

7. Simplified 94.2%

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

Alternative 5: 95.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\mathsf{log1p}\left(\frac{\frac{4}{\pi}}{f} + -1\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* -4.0 (/ (log1p (+ (/ (/ 4.0 PI) f) -1.0)) PI)))

C

double code(double f) {
	return -4.0 * (log1p((((4.0 / ((double) M_PI)) / f) + -1.0)) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log1p((((4.0 / Math.PI) / f) + -1.0)) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log1p((((4.0 / math.pi) / f) + -1.0)) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log1p(Float64(Float64(Float64(4.0 / pi) / f) + -1.0)) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[1 + N[(N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.4%

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

3. Simplified 98.0%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 94.2%

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

   1. mul-1-neg 94.2%

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

   2. unsub-neg 94.2%

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

7. Simplified 94.2%

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

   1. log1p-expm1-u 94.2%

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

   2. expm1-undefine 94.2%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{4}{\pi}\right) - \log f} - 1}\right)}{\pi} \]

   3. diff-log 94.1%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(e^{\color{blue}{\log \left(\frac{\frac{4}{\pi}}{f}\right)}} - 1\right)}{\pi} \]

   4. add-exp-log 94.1%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\frac{\frac{4}{\pi}}{f}} - 1\right)}{\pi} \]

9. Applied egg-rr 94.1%

   \[\leadsto -4 \cdot \frac{\color{blue}{\mathsf{log1p}\left(\frac{\frac{4}{\pi}}{f} - 1\right)}}{\pi} \]

10. Final simplification 94.1%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{\frac{4}{\pi}}{f} + -1\right)}{\pi} \]
11. Add Preprocessing

Alternative 6: 95.9% accurate, 4.9× 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 5.4%

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

3. Simplified 98.0%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 94.2%

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

   1. mul-1-neg 94.2%

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

   2. unsub-neg 94.2%

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

7. Simplified 94.2%

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

   1. diff-log 94.1%

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

9. Applied egg-rr 94.1%

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

Alternative 7: 4.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ -0.08333333333333333 \cdot \left(\pi \cdot {f}^{2}\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* -0.08333333333333333 (* PI (pow f 2.0))))

C

double code(double f) {
	return -0.08333333333333333 * (((double) M_PI) * pow(f, 2.0));
}

Java

public static double code(double f) {
	return -0.08333333333333333 * (Math.PI * Math.pow(f, 2.0));
}

Python

def code(f):
	return -0.08333333333333333 * (math.pi * math.pow(f, 2.0))

Julia

function code(f)
	return Float64(-0.08333333333333333 * Float64(pi * (f ^ 2.0)))
end

MATLAB

function tmp = code(f)
	tmp = -0.08333333333333333 * (pi * (f ^ 2.0));
end

Wolfram

code[f_] := N[(-0.08333333333333333 * N[(Pi * N[Power[f, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.4%

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

3. Simplified 98.0%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 94.4%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + 0.25 \cdot \left({f}^{2} \cdot \left(\pi \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]
6. Step-by-step derivation

   1. +-commutative 94.4%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(0.25 \cdot \left({f}^{2} \cdot \left(\pi \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)\right) + -1 \cdot \log f\right)}\right) \cdot \frac{-4}{\pi} \]

   2. mul-1-neg 94.4%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \left(0.25 \cdot \left({f}^{2} \cdot \left(\pi \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)\right) + \color{blue}{\left(-\log f\right)}\right)\right) \cdot \frac{-4}{\pi} \]

7. Simplified 94.4%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(0.25 \cdot \left({f}^{2} \cdot \left(0.08333333333333333 \cdot {\pi}^{2}\right)\right) - \log f\right)\right)} \cdot \frac{-4}{\pi} \]

8. Taylor expanded in f around inf 4.2%

   \[\leadsto \color{blue}{-0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)} \]

9. Final simplification 4.2%

   \[\leadsto -0.08333333333333333 \cdot \left(\pi \cdot {f}^{2}\right) \]
10. Add Preprocessing

Alternative 8: 0.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\mathsf{log1p}\left(-1\right)}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* -4.0 (/ (log1p -1.0) PI)))

C

double code(double f) {
	return -4.0 * (log1p(-1.0) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log1p(-1.0) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log1p(-1.0) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log1p(-1.0) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[1 + -1.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.4%

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

3. Simplified 98.0%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 94.2%

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

   1. mul-1-neg 94.2%

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

   2. unsub-neg 94.2%

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

7. Simplified 94.2%

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

   1. log1p-expm1-u 94.2%

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

   2. expm1-undefine 94.2%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{4}{\pi}\right) - \log f} - 1}\right)}{\pi} \]

   3. diff-log 94.1%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(e^{\color{blue}{\log \left(\frac{\frac{4}{\pi}}{f}\right)}} - 1\right)}{\pi} \]

   4. add-exp-log 94.1%

      \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\frac{\frac{4}{\pi}}{f}} - 1\right)}{\pi} \]

9. Applied egg-rr 94.1%

   \[\leadsto -4 \cdot \frac{\color{blue}{\mathsf{log1p}\left(\frac{\frac{4}{\pi}}{f} - 1\right)}}{\pi} \]

10. Taylor expanded in f around inf 0.7%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{-1}\right)}{\pi} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024101 
(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)))))))))