VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.3% → 99.0%

Time: 17.0s

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: 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: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\mathsf{log1p}\left(\left(-1 + \frac{-1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot -0.5\right)}\right) + \frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log1p
    (+
     (+ -1.0 (/ -1.0 (expm1 (* (* f PI) -0.5))))
     (/ 1.0 (expm1 (* f (* PI 0.5))))))
   PI)))

C

double code(double f) {
	return -4.0 * (log1p(((-1.0 + (-1.0 / expm1(((f * ((double) M_PI)) * -0.5)))) + (1.0 / expm1((f * (((double) M_PI) * 0.5)))))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log1p(((-1.0 + (-1.0 / Math.expm1(((f * Math.PI) * -0.5)))) + (1.0 / Math.expm1((f * (Math.PI * 0.5)))))) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log1p(((-1.0 + (-1.0 / math.expm1(((f * math.pi) * -0.5)))) + (1.0 / math.expm1((f * (math.pi * 0.5)))))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log1p(Float64(Float64(-1.0 + Float64(-1.0 / expm1(Float64(Float64(f * pi) * -0.5)))) + Float64(1.0 / expm1(Float64(f * Float64(pi * 0.5)))))) / pi))
end

Wolfram

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

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

5. Taylor expanded in f around inf 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

7. Simplified 99.1%

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

   1. log1p-expm1-u 99.1%

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

   2. expm1-undefine 99.1%

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

   3. add-exp-log 99.1%

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

   4. *-commutative 99.1%

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

   5. *-commutative 99.1%

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

   6. associate-*l* 99.1%

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

9. Applied egg-rr 99.1%

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

    1. associate--l+ 99.1%

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

    2. *-commutative 99.1%

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

    3. *-commutative 99.1%

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

    4. *-commutative 99.1%

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

11. Simplified 99.1%

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

12. Final simplification 99.1%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(-1 + \frac{-1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot -0.5\right)}\right) + \frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)}\right)}{\pi} \]
13. Add Preprocessing

Alternative 2: 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%

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

5. Taylor expanded in f around inf 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

7. Simplified 99.1%

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

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 3: 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%

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

5. Taylor expanded in f around 0 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%

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

   6. metadata-eval 97.4%

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

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 4: 96.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\mathsf{log1p}\left(\left(-1 + \frac{-1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot -0.5\right)}\right) + \frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)}{f}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log1p
    (+
     (+ -1.0 (/ -1.0 (expm1 (* (* f PI) -0.5))))
     (/
      (-
       (* 2.0 (/ 1.0 PI))
       (* f (+ 0.5 (* f (+ (* PI -0.125) (* PI 0.08333333333333333))))))
      f)))
   PI)))

C

double code(double f) {
	return -4.0 * (log1p(((-1.0 + (-1.0 / expm1(((f * ((double) M_PI)) * -0.5)))) + (((2.0 * (1.0 / ((double) M_PI))) - (f * (0.5 + (f * ((((double) M_PI) * -0.125) + (((double) M_PI) * 0.08333333333333333)))))) / f))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log1p(((-1.0 + (-1.0 / Math.expm1(((f * Math.PI) * -0.5)))) + (((2.0 * (1.0 / Math.PI)) - (f * (0.5 + (f * ((Math.PI * -0.125) + (Math.PI * 0.08333333333333333)))))) / f))) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log1p(((-1.0 + (-1.0 / math.expm1(((f * math.pi) * -0.5)))) + (((2.0 * (1.0 / math.pi)) - (f * (0.5 + (f * ((math.pi * -0.125) + (math.pi * 0.08333333333333333)))))) / f))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log1p(Float64(Float64(-1.0 + Float64(-1.0 / expm1(Float64(Float64(f * pi) * -0.5)))) + Float64(Float64(Float64(2.0 * Float64(1.0 / pi)) - Float64(f * Float64(0.5 + Float64(f * Float64(Float64(pi * -0.125) + Float64(pi * 0.08333333333333333)))))) / f))) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[1 + N[(N[(-1.0 + N[(-1.0 / N[(Exp[N[(N[(f * Pi), $MachinePrecision] * -0.5), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] - N[(f * N[(0.5 + N[(f * N[(N[(Pi * -0.125), $MachinePrecision] + N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $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%

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

5. Taylor expanded in f around inf 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

7. Simplified 99.1%

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

   1. log1p-expm1-u 99.1%

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

   2. expm1-undefine 99.1%

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

   3. add-exp-log 99.1%

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

   4. *-commutative 99.1%

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

   5. *-commutative 99.1%

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

   6. associate-*l* 99.1%

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

9. Applied egg-rr 99.1%

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

    1. associate--l+ 99.1%

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

    2. *-commutative 99.1%

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

    3. *-commutative 99.1%

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

    4. *-commutative 99.1%

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

11. Simplified 99.1%

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

    1. add-cbrt-cube 99.1%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}} \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot -0.5\right)} - 1\right)\right)}{\pi} \]

    2. pow3 99.1%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\sqrt[3]{\color{blue}{{\pi}^{3}}} \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot -0.5\right)} - 1\right)\right)}{\pi} \]

13. Applied egg-rr 99.1%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot 0.5\right)\right)} + \left(\frac{-1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot -0.5\right)} - 1\right)\right)}{\pi} \]

14. Taylor expanded in f around 0 97.1%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\frac{f \cdot \left(-1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}} + \left(\frac{-1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot -0.5\right)} - 1\right)\right)}{\pi} \]

15. Final simplification 97.1%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\left(-1 + \frac{-1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot -0.5\right)}\right) + \frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)}{f}\right)}{\pi} \]
16. Add Preprocessing

Alternative 5: 96.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(-1 + f \cdot \left(\left(\pi \cdot -0.08333333333333333 + \pi \cdot 0.125\right) - \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log1p
    (/
     (+
      (*
       f
       (+
        -1.0
        (*
         f
         (-
          (+ (* PI -0.08333333333333333) (* PI 0.125))
          (+ (* PI -0.125) (* PI 0.08333333333333333))))))
      (* 4.0 (/ 1.0 PI)))
     f))
   PI)))

C

double code(double f) {
	return -4.0 * (log1p((((f * (-1.0 + (f * (((((double) M_PI) * -0.08333333333333333) + (((double) M_PI) * 0.125)) - ((((double) M_PI) * -0.125) + (((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.log1p((((f * (-1.0 + (f * (((Math.PI * -0.08333333333333333) + (Math.PI * 0.125)) - ((Math.PI * -0.125) + (Math.PI * 0.08333333333333333)))))) + (4.0 * (1.0 / Math.PI))) / f)) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log1p((((f * (-1.0 + (f * (((math.pi * -0.08333333333333333) + (math.pi * 0.125)) - ((math.pi * -0.125) + (math.pi * 0.08333333333333333)))))) + (4.0 * (1.0 / math.pi))) / f)) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log1p(Float64(Float64(Float64(f * Float64(-1.0 + Float64(f * Float64(Float64(Float64(pi * -0.08333333333333333) + Float64(pi * 0.125)) - Float64(Float64(pi * -0.125) + Float64(pi * 0.08333333333333333)))))) + Float64(4.0 * Float64(1.0 / pi))) / f)) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[1 + N[(N[(N[(f * N[(-1.0 + N[(f * N[(N[(N[(Pi * -0.08333333333333333), $MachinePrecision] + N[(Pi * 0.125), $MachinePrecision]), $MachinePrecision] - N[(N[(Pi * -0.125), $MachinePrecision] + N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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%

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

5. Taylor expanded in f around inf 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

7. Simplified 99.1%

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

   1. log1p-expm1-u 99.1%

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

   2. expm1-undefine 99.1%

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

   3. add-exp-log 99.1%

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

   4. *-commutative 99.1%

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

   5. *-commutative 99.1%

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

   6. associate-*l* 99.1%

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

9. Applied egg-rr 99.1%

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

    1. associate--l+ 99.1%

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

    2. *-commutative 99.1%

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

    3. *-commutative 99.1%

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

    4. *-commutative 99.1%

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

11. Simplified 99.1%

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

12. Taylor expanded in f around 0 97.1%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\frac{f \cdot \left(f \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right) - 1\right) + 4 \cdot \frac{1}{\pi}}{f}}\right)}{\pi} \]

13. Final simplification 97.1%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(-1 + f \cdot \left(\left(\pi \cdot -0.08333333333333333 + \pi \cdot 0.125\right) - \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]
14. Add Preprocessing

Alternative 6: 95.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in f around 0 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. Step-by-step derivation

   1. add-exp-log 95.7%

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

   2. diff-log 95.4%

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

9. Applied egg-rr 95.4%

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

    1. rem-exp-log 96.4%

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

    2. clear-num 96.4%

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

    3. log-div 96.8%

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

    4. metadata-eval 96.8%

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

    5. div-inv 96.8%

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

    6. clear-num 96.8%

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

    7. div-inv 96.8%

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

    8. metadata-eval 96.8%

       \[\leadsto \left(0 - \log \left(f \cdot \left(\pi \cdot \color{blue}{0.25}\right)\right)\right) \cdot \frac{-4}{\pi} \]

11. Applied egg-rr 96.8%

    \[\leadsto \color{blue}{\left(0 - \log \left(f \cdot \left(\pi \cdot 0.25\right)\right)\right)} \cdot \frac{-4}{\pi} \]
12. Step-by-step derivation

    1. neg-sub0 96.8%

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

    2. associate-*r* 96.8%

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

    3. *-commutative 96.8%

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

13. Simplified 96.8%

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

14. Final simplification 96.8%

    \[\leadsto \log \left(\left(f \cdot \pi\right) \cdot 0.25\right) \cdot \frac{-4}{-\pi} \]
15. 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%

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

5. Taylor expanded in f around inf 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

7. Simplified 99.1%

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

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: 3.1% accurate, 5.3× speedup

Math

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

FPCore

(FPCore (f) :precision binary64 (log 0.0))

C

double code(double f) {
	return log(0.0);
}

Fortran

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

Java

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

Python

def code(f):
	return math.log(0.0)

Julia

function code(f)
	return log(0.0)
end

MATLAB

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

Wolfram

code[f_] := N[Log[0.0], $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%

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

6. Applied egg-rr 0.7%

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

   1. +-inverses 0.7%

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

8. Simplified 0.7%

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

   1. add-log-exp 0.7%

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

   2. exp-to-pow 0.7%

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

10. Applied egg-rr 0.7%

    \[\leadsto \color{blue}{\log \left({0}^{\left(\frac{-4}{\pi}\right)}\right)} \]
11. Step-by-step derivation

    1. pow-base-0 3.1%

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

12. Simplified 3.1%

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