VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 99.0%

Time: 19.2s

Alternatives: 10

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: 6.9% 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

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

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around inf 7.9%

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

   1. associate-*r/ 7.9%

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

7. Simplified 99.2%

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

   1. log1p-expm1-u 99.2%

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

   2. expm1-undefine 99.2%

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

   3. add-exp-log 99.2%

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

9. Applied egg-rr 99.2%

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

    1. associate--l+ 99.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in f around inf 7.9%

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

    1. associate--r+ 7.9%

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

    2. expm1-define 8.0%

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

    3. *-commutative 8.0%

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

    4. associate-*r* 8.0%

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

    5. sub-neg 8.0%

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

    6. sub-neg 8.0%

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

    7. metadata-eval 8.0%

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

    8. +-commutative 8.0%

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

    9. distribute-neg-frac 8.0%

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

    10. metadata-eval 8.0%

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

    11. expm1-define 99.2%

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

    12. *-commutative 99.2%

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

    13. *-commutative 99.2%

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

14. Simplified 99.2%

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

15. Final simplification 99.2%

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

Alternative 2: 99.0% accurate, 1.7× speedup

Math

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

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

Derivation

1. Initial program 8.0%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around inf 7.9%

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

   1. associate-*r/ 7.9%

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

7. Simplified 99.2%

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

Alternative 3: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 4: 98.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\\ \mathbf{if}\;f \leq 2.35:\\ \;\;\;\;\frac{-4 \cdot \log \left(t\_0 + \frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ -1.0 (expm1 (* PI (* f -0.5))))))
   (if (<= f 2.35)
     (/
      (*
       -4.0
       (log
        (+
         t_0
         (/
          (-
           (* 2.0 (/ 1.0 PI))
           (* f (+ 0.5 (* f (+ (* PI -0.125) (* PI 0.08333333333333333))))))
          f))))
      PI)
     (* (/ -4.0 PI) (log t_0)))))

C

double code(double f) {
	double t_0 = -1.0 / expm1((((double) M_PI) * (f * -0.5)));
	double tmp;
	if (f <= 2.35) {
		tmp = (-4.0 * log((t_0 + (((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);
	} else {
		tmp = (-4.0 / ((double) M_PI)) * log(t_0);
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = -1.0 / Math.expm1((Math.PI * (f * -0.5)));
	double tmp;
	if (f <= 2.35) {
		tmp = (-4.0 * Math.log((t_0 + (((2.0 * (1.0 / Math.PI)) - (f * (0.5 + (f * ((Math.PI * -0.125) + (Math.PI * 0.08333333333333333)))))) / f)))) / Math.PI;
	} else {
		tmp = (-4.0 / Math.PI) * Math.log(t_0);
	}
	return tmp;
}

Python

def code(f):
	t_0 = -1.0 / math.expm1((math.pi * (f * -0.5)))
	tmp = 0
	if f <= 2.35:
		tmp = (-4.0 * math.log((t_0 + (((2.0 * (1.0 / math.pi)) - (f * (0.5 + (f * ((math.pi * -0.125) + (math.pi * 0.08333333333333333)))))) / f)))) / math.pi
	else:
		tmp = (-4.0 / math.pi) * math.log(t_0)
	return tmp

Julia

function code(f)
	t_0 = Float64(-1.0 / expm1(Float64(pi * Float64(f * -0.5))))
	tmp = 0.0
	if (f <= 2.35)
		tmp = Float64(Float64(-4.0 * log(Float64(t_0 + 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);
	else
		tmp = Float64(Float64(-4.0 / pi) * log(t_0));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[(-1.0 / N[(Exp[N[(Pi * N[(f * -0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[f, 2.35], N[(N[(-4.0 * N[Log[N[(t$95$0 + 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]), $MachinePrecision] / Pi), $MachinePrecision], N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if f < 2.35000000000000009

   1. Initial program 7.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in f around inf 4.4%

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

      1. associate-*r/ 4.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in f around 0 99.0%

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

   if 2.35000000000000009 < f

   1. Initial program 23.4%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in f around 0 5.0%

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

   6. Taylor expanded in f around inf 83.8%

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

      1. distribute-neg-frac 83.8%

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

      2. metadata-eval 83.8%

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

      3. *-commutative 83.8%

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

      4. *-commutative 83.8%

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

      5. associate-*r* 83.8%

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

      6. expm1-undefine 83.8%

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

   8. Simplified 83.8%

      \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)} \cdot \frac{-4}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 2.35:\\ \;\;\;\;\frac{-4 \cdot \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} + \frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 2.1:\\ \;\;\;\;\frac{-4 \cdot \mathsf{log1p}\left(\frac{f \cdot \left(-1 + f \cdot \left(\left(\pi \cdot -0.08333333333333333 + \pi \cdot 0.125\right) - \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)\right) + \frac{1}{\pi} \cdot 4}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 2.1)
   (/
    (*
     -4.0
     (log1p
      (/
       (+
        (*
         f
         (+
          -1.0
          (*
           f
           (-
            (+ (* PI -0.08333333333333333) (* PI 0.125))
            (+ (* PI -0.125) (* PI 0.08333333333333333))))))
        (* (/ 1.0 PI) 4.0))
       f)))
    PI)
   (* (/ -4.0 PI) (log (/ -1.0 (expm1 (* PI (* f -0.5))))))))

C

double code(double f) {
	double tmp;
	if (f <= 2.1) {
		tmp = (-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)))))) + ((1.0 / ((double) M_PI)) * 4.0)) / f))) / ((double) M_PI);
	} else {
		tmp = (-4.0 / ((double) M_PI)) * log((-1.0 / expm1((((double) M_PI) * (f * -0.5)))));
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 2.1) {
		tmp = (-4.0 * Math.log1p((((f * (-1.0 + (f * (((Math.PI * -0.08333333333333333) + (Math.PI * 0.125)) - ((Math.PI * -0.125) + (Math.PI * 0.08333333333333333)))))) + ((1.0 / Math.PI) * 4.0)) / f))) / Math.PI;
	} else {
		tmp = (-4.0 / Math.PI) * Math.log((-1.0 / Math.expm1((Math.PI * (f * -0.5)))));
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 2.1:
		tmp = (-4.0 * math.log1p((((f * (-1.0 + (f * (((math.pi * -0.08333333333333333) + (math.pi * 0.125)) - ((math.pi * -0.125) + (math.pi * 0.08333333333333333)))))) + ((1.0 / math.pi) * 4.0)) / f))) / math.pi
	else:
		tmp = (-4.0 / math.pi) * math.log((-1.0 / math.expm1((math.pi * (f * -0.5)))))
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 2.1)
		tmp = Float64(Float64(-4.0 * 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(Float64(1.0 / pi) * 4.0)) / f))) / pi);
	else
		tmp = Float64(Float64(-4.0 / pi) * log(Float64(-1.0 / expm1(Float64(pi * Float64(f * -0.5))))));
	end
	return tmp
end

Wolfram

code[f_] := If[LessEqual[f, 2.1], N[(N[(-4.0 * 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[(N[(1.0 / Pi), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(-1.0 / N[(Exp[N[(Pi * N[(f * -0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < 2.10000000000000009

   1. Initial program 7.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in f around inf 4.4%

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

      1. associate-*r/ 4.4%

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

   7. Simplified 99.4%

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

      1. log1p-expm1-u 99.4%

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

      2. expm1-undefine 99.4%

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

      3. add-exp-log 99.4%

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

   9. Applied egg-rr 99.4%

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

       1. associate--l+ 99.4%

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

   11. Simplified 99.4%

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

   12. Taylor expanded in f around 0 98.9%

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

   if 2.10000000000000009 < f

   1. Initial program 23.4%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in f around 0 5.0%

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

   6. Taylor expanded in f around inf 83.8%

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

      1. distribute-neg-frac 83.8%

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

      2. metadata-eval 83.8%

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

      3. *-commutative 83.8%

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

      4. *-commutative 83.8%

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

      5. associate-*r* 83.8%

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

      6. expm1-undefine 83.8%

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

   8. Simplified 83.8%

      \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)} \cdot \frac{-4}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 2.1:\\ \;\;\;\;\frac{-4 \cdot \mathsf{log1p}\left(\frac{f \cdot \left(-1 + f \cdot \left(\left(\pi \cdot -0.08333333333333333 + \pi \cdot 0.125\right) - \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)\right) + \frac{1}{\pi} \cdot 4}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.4% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (/
  (*
   -4.0
   (log1p
    (/
     (+
      (*
       f
       (+
        -1.0
        (*
         f
         (-
          (+ (* PI -0.08333333333333333) (* PI 0.125))
          (+ (* PI -0.125) (* PI 0.08333333333333333))))))
      (* (/ 1.0 PI) 4.0))
     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)))))) + ((1.0 / ((double) M_PI)) * 4.0)) / 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)))))) + ((1.0 / Math.PI) * 4.0)) / 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)))))) + ((1.0 / math.pi) * 4.0)) / f))) / math.pi

Julia

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around inf 7.9%

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

   1. associate-*r/ 7.9%

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

7. Simplified 99.2%

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

   1. log1p-expm1-u 99.2%

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

   2. expm1-undefine 99.2%

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

   3. add-exp-log 99.2%

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

9. Applied egg-rr 99.2%

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

    1. associate--l+ 99.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in f around 0 95.3%

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

13. Final simplification 95.3%

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

Alternative 7: 95.8% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around 0 94.7%

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

   1. associate-*r/ 94.7%

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

7. Applied egg-rr 94.7%

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

8. Final simplification 94.7%

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

Alternative 8: 95.7% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around 0 94.7%

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

6. Final simplification 94.7%

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

Alternative 9: 5.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{-16}{f \cdot {\pi}^{2}} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ -16.0 (* f (pow PI 2.0))))

C

double code(double f) {
	return -16.0 / (f * pow(((double) M_PI), 2.0));
}

Java

public static double code(double f) {
	return -16.0 / (f * Math.pow(Math.PI, 2.0));
}

Python

def code(f):
	return -16.0 / (f * math.pow(math.pi, 2.0))

Julia

function code(f)
	return Float64(-16.0 / Float64(f * (pi ^ 2.0)))
end

MATLAB

function tmp = code(f)
	tmp = -16.0 / (f * (pi ^ 2.0));
end

Wolfram

code[f_] := N[(-16.0 / N[(f * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.0%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around inf 7.9%

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

   1. associate-*r/ 7.9%

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

7. Simplified 99.2%

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

   1. log1p-expm1-u 99.2%

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

   2. expm1-undefine 99.2%

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

   3. add-exp-log 99.2%

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

9. Applied egg-rr 99.2%

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

    1. associate--l+ 99.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in f around 0 93.7%

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

    1. *-commutative 93.7%

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

14. Simplified 93.7%

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

15. Taylor expanded in f around inf 5.4%

    \[\leadsto \color{blue}{\frac{-16}{f \cdot {\pi}^{2}}} \]
16. Add Preprocessing

Alternative 10: 5.4% accurate, 76.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-16}{f \cdot \pi}}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (/ -16.0 (* f PI)) PI))

C

double code(double f) {
	return (-16.0 / (f * ((double) M_PI))) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (-16.0 / (f * Math.PI)) / Math.PI;
}

Python

def code(f):
	return (-16.0 / (f * math.pi)) / math.pi

Julia

function code(f)
	return Float64(Float64(-16.0 / Float64(f * pi)) / pi)
end

MATLAB

function tmp = code(f)
	tmp = (-16.0 / (f * pi)) / pi;
end

Wolfram

code[f_] := N[(N[(-16.0 / N[(f * Pi), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 8.0%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around inf 7.9%

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

   1. associate-*r/ 7.9%

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

7. Simplified 99.2%

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

   1. log1p-expm1-u 99.2%

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

   2. expm1-undefine 99.2%

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

   3. add-exp-log 99.2%

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

9. Applied egg-rr 99.2%

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

    1. associate--l+ 99.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in f around 0 93.7%

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

    1. *-commutative 93.7%

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

14. Simplified 93.7%

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

15. Taylor expanded in f around inf 5.4%

    \[\leadsto \frac{\color{blue}{\frac{-16}{f \cdot \pi}}}{\pi} \]
16. Step-by-step derivation

    1. *-commutative 5.4%

       \[\leadsto \frac{\frac{-16}{\color{blue}{\pi \cdot f}}}{\pi} \]

17. Simplified 5.4%

    \[\leadsto \frac{\color{blue}{\frac{-16}{\pi \cdot f}}}{\pi} \]

18. Final simplification 5.4%

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

Reproduce

herbie shell --seed 2024150 
(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)))))))))