VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.3% → 98.9%

Time: 18.2s

Alternatives: 9

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: 7.3% 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: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Python

def code(f):
	return -4.0 * (math.log(((1.0 / math.expm1((f * (math.pi * 0.5)))) + (-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(f * Float64(pi * 0.5)))) + 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[(f * N[(Pi * 0.5), $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 8.5%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around inf 7.3%

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

   1. expm1-define 7.5%

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

   2. *-commutative 7.5%

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

   3. associate-*l* 7.5%

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

   4. expm1-define 99.1%

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

   5. associate-*r* 99.1%

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

   6. *-commutative 99.1%

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

   7. *-commutative 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 2: 96.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(f)
	tmp = (-4.0 * ((log((4.0 / pi)) - log(f)) / pi)) - ((f ^ 2.0) * (pi * 0.08333333333333333));
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[(N[Power[f, 2.0], $MachinePrecision] * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.5%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around 0 97.4%

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

   1. mul-1-neg 97.4%

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

   2. unsub-neg 97.4%

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

   3. mul-1-neg 97.4%

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

   4. unsub-neg 97.4%

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

   5. distribute-rgt-out 97.4%

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

   6. metadata-eval 97.4%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} - {f}^{2} \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \pi \cdot \color{blue}{-0.041666666666666664}\right) \]

7. Simplified 97.4%

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

Alternative 3: 96.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{{f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log
    (/ (+ (* (pow f 2.0) (* PI 0.08333333333333333)) (* 4.0 (/ 1.0 PI))) f))
   PI)))

C

double code(double f) {
	return -4.0 * (log((((pow(f, 2.0) * (((double) M_PI) * 0.08333333333333333)) + (4.0 * (1.0 / ((double) M_PI)))) / f)) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log((((Math.pow(f, 2.0) * (Math.PI * 0.08333333333333333)) + (4.0 * (1.0 / Math.PI))) / f)) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log((((math.pow(f, 2.0) * (math.pi * 0.08333333333333333)) + (4.0 * (1.0 / math.pi))) / f)) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log(Float64(Float64(Float64((f ^ 2.0) * Float64(pi * 0.08333333333333333)) + Float64(4.0 * Float64(1.0 / pi))) / f)) / pi))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 * (log(((((f ^ 2.0) * (pi * 0.08333333333333333)) + (4.0 * (1.0 / pi))) / f)) / pi);
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(N[(N[(N[Power[f, 2.0], $MachinePrecision] * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.5%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around inf 7.3%

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

   1. expm1-define 7.5%

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

   2. *-commutative 7.5%

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

   3. associate-*l* 7.5%

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

   4. expm1-define 99.1%

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

   5. associate-*r* 99.1%

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

   6. *-commutative 99.1%

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

   7. *-commutative 99.1%

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

7. Simplified 99.1%

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

8. Taylor expanded in f around 0 97.1%

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

   1. pow1 97.1%

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

   2. distribute-rgt-out 97.1%

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

   3. metadata-eval 97.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{{\left({f}^{2} \cdot \left(\pi \cdot \color{blue}{0.041666666666666664} - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)}^{1} + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

   4. distribute-rgt-out 97.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{{\left({f}^{2} \cdot \left(\pi \cdot 0.041666666666666664 - \color{blue}{\pi \cdot \left(-0.125 + 0.08333333333333333\right)}\right)\right)}^{1} + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

   5. metadata-eval 97.1%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{{\left({f}^{2} \cdot \left(\pi \cdot 0.041666666666666664 - \pi \cdot \color{blue}{-0.041666666666666664}\right)\right)}^{1} + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

10. Applied egg-rr 97.1%

    \[\leadsto -4 \cdot \frac{\log \left(\frac{\color{blue}{{\left({f}^{2} \cdot \left(\pi \cdot 0.041666666666666664 - \pi \cdot -0.041666666666666664\right)\right)}^{1}} + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]
11. Step-by-step derivation

    1. unpow1 97.1%

       \[\leadsto -4 \cdot \frac{\log \left(\frac{\color{blue}{{f}^{2} \cdot \left(\pi \cdot 0.041666666666666664 - \pi \cdot -0.041666666666666664\right)} + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

    2. distribute-lft-out-- 97.1%

       \[\leadsto -4 \cdot \frac{\log \left(\frac{{f}^{2} \cdot \color{blue}{\left(\pi \cdot \left(0.041666666666666664 - -0.041666666666666664\right)\right)} + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

    3. metadata-eval 97.1%

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

12. Simplified 97.1%

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

Alternative 4: 95.5% accurate, 2.5× speedup

Math

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

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

Derivation

1. Initial program 8.5%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around 0 97.0%

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

   1. associate-*r/ 97.0%

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

   2. mul-1-neg 97.0%

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

   3. unsub-neg 97.0%

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

7. Simplified 97.0%

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

Alternative 5: 95.5% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.5%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around 0 96.9%

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

   1. mul-1-neg 96.9%

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

   2. unsub-neg 96.9%

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

7. Simplified 96.9%

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

Alternative 6: 95.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(N[Log[N[(0.25 * N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / (-Pi)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.5%

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

3. Simplified 98.9%

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

6. Applied egg-rr 0.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right) \cdot \frac{-4}{\pi}\right)} - 1} \]
7. Step-by-step derivation

   1. sub-neg 0.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right) \cdot \frac{-4}{\pi}\right)} + \left(-1\right)} \]

   2. metadata-eval 0.1%

      \[\leadsto e^{\mathsf{log1p}\left(\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right) \cdot \frac{-4}{\pi}\right)} + \color{blue}{-1} \]

   3. +-commutative 0.1%

      \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right) \cdot \frac{-4}{\pi}\right)}} \]

   4. log1p-undefine 0.1%

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

   5. rem-exp-log 94.8%

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

   6. associate-+r+ 94.8%

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

   7. metadata-eval 94.8%

      \[\leadsto \color{blue}{0} + \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   8. mul0-lft 94.8%

      \[\leadsto \color{blue}{0 \cdot \frac{-4}{\pi}} + \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right) \cdot \frac{-4}{\pi} \]

8. Simplified 94.8%

   \[\leadsto \color{blue}{\log \left(\frac{-2}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right) \cdot \frac{-4}{\pi}} \]

9. Taylor expanded in f around 0 94.8%

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

    1. associate-*r/ 94.8%

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

    2. metadata-eval 94.8%

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

11. Simplified 94.8%

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

    1. clear-num 94.8%

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

    2. log-div 95.2%

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

    3. 1-exp 95.2%

       \[\leadsto \left(\log \color{blue}{\left(e^{0}\right)} - \log \left(\frac{f}{f + \frac{4}{\pi}}\right)\right) \cdot \frac{-4}{\pi} \]

    4. mul0-lft 95.2%

       \[\leadsto \left(\log \left(e^{\color{blue}{0 \cdot \frac{-4}{\pi}}}\right) - \log \left(\frac{f}{f + \frac{4}{\pi}}\right)\right) \cdot \frac{-4}{\pi} \]

    5. add-log-exp 95.2%

       \[\leadsto \left(\color{blue}{0 \cdot \frac{-4}{\pi}} - \log \left(\frac{f}{f + \frac{4}{\pi}}\right)\right) \cdot \frac{-4}{\pi} \]

    6. mul0-lft 95.2%

       \[\leadsto \left(\color{blue}{0} - \log \left(\frac{f}{f + \frac{4}{\pi}}\right)\right) \cdot \frac{-4}{\pi} \]

13. Applied egg-rr 95.2%

    \[\leadsto \color{blue}{\left(0 - \log \left(\frac{f}{f + \frac{4}{\pi}}\right)\right)} \cdot \frac{-4}{\pi} \]
14. Step-by-step derivation

    1. neg-sub0 95.2%

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

    2. +-commutative 95.2%

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

15. Simplified 95.2%

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

16. Taylor expanded in f around 0 96.8%

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

    1. *-commutative 96.8%

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

18. Simplified 96.8%

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

19. Final simplification 96.8%

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

Alternative 7: 95.5% accurate, 4.9× speedup

Math

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

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

Derivation

1. Initial program 8.5%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around inf 7.3%

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

   1. expm1-define 7.5%

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

   2. *-commutative 7.5%

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

   3. associate-*l* 7.5%

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

   4. expm1-define 99.1%

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

   5. associate-*r* 99.1%

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

   6. *-commutative 99.1%

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

   7. *-commutative 99.1%

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

7. Simplified 99.1%

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

8. Taylor expanded in f around 0 96.6%

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

   1. *-commutative 96.6%

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

10. Simplified 96.6%

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

11. Final simplification 96.6%

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

Alternative 8: 5.5% accurate, 59.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.5%

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

3. Simplified 98.9%

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

6. Applied egg-rr 0.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right) \cdot \frac{-4}{\pi}\right)} - 1} \]
7. Step-by-step derivation

   1. sub-neg 0.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right) \cdot \frac{-4}{\pi}\right)} + \left(-1\right)} \]

   2. metadata-eval 0.1%

      \[\leadsto e^{\mathsf{log1p}\left(\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right) \cdot \frac{-4}{\pi}\right)} + \color{blue}{-1} \]

   3. +-commutative 0.1%

      \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right) \cdot \frac{-4}{\pi}\right)}} \]

   4. log1p-undefine 0.1%

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

   5. rem-exp-log 94.8%

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

   6. associate-+r+ 94.8%

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

   7. metadata-eval 94.8%

      \[\leadsto \color{blue}{0} + \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   8. mul0-lft 94.8%

      \[\leadsto \color{blue}{0 \cdot \frac{-4}{\pi}} + \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right) \cdot \frac{-4}{\pi} \]

8. Simplified 94.8%

   \[\leadsto \color{blue}{\log \left(\frac{-2}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right) \cdot \frac{-4}{\pi}} \]

9. Taylor expanded in f around 0 94.8%

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

    1. associate-*r/ 94.8%

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

    2. metadata-eval 94.8%

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

11. Simplified 94.8%

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

12. Taylor expanded in f around inf 5.6%

    \[\leadsto \color{blue}{\frac{4}{f \cdot \pi}} \cdot \frac{-4}{\pi} \]
13. Step-by-step derivation

    1. *-commutative 5.6%

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

14. Simplified 5.6%

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

15. Final simplification 5.6%

    \[\leadsto \frac{-4}{\pi} \cdot \frac{4}{f \cdot \pi} \]
16. Add Preprocessing

Alternative 9: 5.3% accurate, 532.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (f) :precision binary64 0.0)

C

double code(double f) {
	return 0.0;
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = 0.0d0
end function

Java

public static double code(double f) {
	return 0.0;
}

Python

def code(f):
	return 0.0

Julia

function code(f)
	return 0.0
end

MATLAB

function tmp = code(f)
	tmp = 0.0;
end

Wolfram

code[f_] := 0.0

Derivation

1. Initial program 8.5%

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

3. Simplified 98.9%

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

6. Applied egg-rr 94.5%

   \[\leadsto \color{blue}{{\left(\sqrt{\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(-0.5 \cdot f\right)\right)}\right)}\right)}^{2}} \cdot \frac{-4}{\pi} \]
7. Step-by-step derivation

   1. unpow2 94.5%

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

   2. add-sqr-sqrt 94.8%

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

   3. flip-+ 0.0%

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

   4. log-div 0.0%

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

8. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{\left(\log \left({\left(\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)\right)}^{-2} - {\left(\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)\right)}^{-2}\right) - \log \left(\frac{0}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\right)} \cdot \frac{-4}{\pi} \]
9. Step-by-step derivation

   1. +-inverses 0.0%

      \[\leadsto \left(\log \color{blue}{0} - \log \left(\frac{0}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\right) \cdot \frac{-4}{\pi} \]

   2. div0 0.0%

      \[\leadsto \left(\log 0 - \log \color{blue}{0}\right) \cdot \frac{-4}{\pi} \]

   3. +-inverses 4.6%

      \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]

10. Simplified 4.6%

    \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]
11. Step-by-step derivation

    1. mul0-lft 4.6%

       \[\leadsto \color{blue}{0} \]

12. Applied egg-rr 4.6%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024149 
(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)))))))))