VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.1% → 99.0%

Time: 18.2s

Alternatives: 9

Speedup: 4.9×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t\_0}\\ t_2 := e^{-t\_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t\_1 + t\_2}{t\_1 - t\_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Initial Program: 7.1% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.9%

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

3. Simplified 98.5%

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

5. Taylor expanded in f around inf 5.5%

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

   1. associate-*r/ 5.5%

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

7. Simplified 98.7%

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

   1. expm1-log1p-u 97.5%

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

   2. expm1-undefine 97.5%

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

9. Applied egg-rr 97.5%

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

    1. expm1-define 97.5%

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

    2. expm1-log1p-u 98.7%

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

    3. log1p-expm1-u 98.7%

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

    4. log1p-undefine 98.7%

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

    5. expm1-undefine 98.7%

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

    6. add-exp-log 98.7%

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

11. Applied egg-rr 98.7%

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

    1. log1p-define 98.7%

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

    2. associate--l+ 98.8%

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

13. Simplified 98.8%

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

    1. add-exp-log 98.8%

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

    2. sub-neg 98.8%

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

    3. metadata-eval 98.8%

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

15. Applied egg-rr 98.8%

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

16. Final simplification 98.8%

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

Alternative 2: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Python

def code(f):
	return (-4.0 * math.log1p(((1.0 / math.expm1((f * (math.pi * 0.5)))) + (-1.0 + (-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 / expm1(Float64(f * Float64(pi * 0.5)))) + Float64(-1.0 + Float64(-1.0 / expm1(Float64(f * Float64(pi * -0.5)))))))) / pi)
end

Wolfram

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

Derivation

1. Initial program 5.9%

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

3. Simplified 98.5%

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

5. Taylor expanded in f around inf 5.5%

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

   1. associate-*r/ 5.5%

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

7. Simplified 98.7%

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

   1. expm1-log1p-u 97.5%

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

   2. expm1-undefine 97.5%

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

9. Applied egg-rr 97.5%

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

    1. expm1-define 97.5%

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

    2. expm1-log1p-u 98.7%

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

    3. log1p-expm1-u 98.7%

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

    4. log1p-undefine 98.7%

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

    5. expm1-undefine 98.7%

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

    6. add-exp-log 98.7%

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

11. Applied egg-rr 98.7%

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

    1. log1p-define 98.7%

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

    2. associate--l+ 98.8%

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

13. Simplified 98.8%

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

14. Final simplification 98.8%

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

Alternative 3: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double f) {
	return (-4.0 * log(((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.log(((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.log(((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 * log(Float64(Float64(1.0 / expm1(Float64(f * Float64(pi * 0.5)))) + Float64(-1.0 / expm1(Float64(f * Float64(pi * -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[(f * N[(Pi * -0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 5.9%

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

3. Simplified 98.5%

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

5. Taylor expanded in f around inf 5.5%

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

   1. associate-*r/ 5.5%

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

7. Simplified 98.7%

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

Alternative 4: 98.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.9%

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

3. Simplified 98.5%

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

5. Final simplification 98.5%

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

Alternative 5: 98.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;\frac{-4 \cdot \mathsf{log1p}\left(\frac{f \cdot \left(-1 + f \cdot \left(\left(\pi \cdot -0.08333333333333333 + \pi \cdot 0.125\right) - \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 225.0)
   (/
    (*
     -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)
   (/ (* -4.0 (log (/ -1.0 (expm1 (* PI (* f -0.5)))))) PI)))

C

double code(double f) {
	double tmp;
	if (f <= 225.0) {
		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)))))) + (4.0 * (1.0 / ((double) M_PI)))) / f))) / ((double) M_PI);
	} else {
		tmp = (-4.0 * log((-1.0 / expm1((((double) M_PI) * (f * -0.5)))))) / ((double) M_PI);
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 225.0) {
		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)))))) + (4.0 * (1.0 / Math.PI))) / f))) / Math.PI;
	} else {
		tmp = (-4.0 * Math.log((-1.0 / Math.expm1((Math.PI * (f * -0.5)))))) / Math.PI;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 225.0:
		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)))))) + (4.0 * (1.0 / math.pi))) / f))) / math.pi
	else:
		tmp = (-4.0 * math.log((-1.0 / math.expm1((math.pi * (f * -0.5)))))) / math.pi
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 225.0)
		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(4.0 * Float64(1.0 / pi))) / f))) / pi);
	else
		tmp = Float64(Float64(-4.0 * log(Float64(-1.0 / expm1(Float64(pi * Float64(f * -0.5)))))) / pi);
	end
	return tmp
end

Wolfram

code[f_] := If[LessEqual[f, 225.0], 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[(4.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], N[(N[(-4.0 * N[Log[N[(-1.0 / N[(Exp[N[(Pi * N[(f * -0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < 225

   1. Initial program 6.1%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in f around inf 3.2%

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

      1. associate-*r/ 3.2%

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

   7. Simplified 98.6%

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

      1. expm1-log1p-u 97.5%

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

      2. expm1-undefine 97.5%

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

   9. Applied egg-rr 97.5%

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

       1. expm1-define 97.5%

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

       2. expm1-log1p-u 98.6%

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

       3. log1p-expm1-u 98.6%

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

       4. log1p-undefine 98.6%

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

       5. expm1-undefine 98.6%

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

       6. add-exp-log 98.6%

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

   11. Applied egg-rr 98.6%

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

       1. log1p-define 98.6%

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

       2. associate--l+ 98.8%

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

   13. Simplified 98.8%

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

   14. Taylor expanded in f around 0 97.6%

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

   if 225 < f

   1. Initial program 0.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in f around 0 3.1%

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

   6. Taylor expanded in f around 0 3.2%

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

      1. *-commutative 3.2%

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

   8. Simplified 3.2%

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

   9. Taylor expanded in f around inf 100.0%

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

       1. associate-*r/ 100.0%

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

       2. distribute-neg-frac 100.0%

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

       3. metadata-eval 100.0%

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

       4. expm1-define 100.0%

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

       5. *-commutative 100.0%

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

       6. *-commutative 100.0%

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

       7. associate-*l* 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;\frac{-4 \cdot \mathsf{log1p}\left(\frac{f \cdot \left(-1 + f \cdot \left(\left(\pi \cdot -0.08333333333333333 + \pi \cdot 0.125\right) - \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.2% accurate, 4.0× speedup

Math

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

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(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(4.0 * Float64(1.0 / pi))) / 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[(4.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 5.9%

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

3. Simplified 98.5%

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

5. Taylor expanded in f around inf 5.5%

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

   1. associate-*r/ 5.5%

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

7. Simplified 98.7%

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

   1. expm1-log1p-u 97.5%

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

   2. expm1-undefine 97.5%

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

9. Applied egg-rr 97.5%

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

    1. expm1-define 97.5%

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

    2. expm1-log1p-u 98.7%

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

    3. log1p-expm1-u 98.7%

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

    4. log1p-undefine 98.7%

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

    5. expm1-undefine 98.7%

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

    6. add-exp-log 98.7%

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

11. Applied egg-rr 98.7%

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

    1. log1p-define 98.7%

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

    2. associate--l+ 98.8%

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

13. Simplified 98.8%

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

14. Taylor expanded in f around 0 95.4%

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

15. Final simplification 95.4%

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

Alternative 7: 95.6% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.9%

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

3. Simplified 98.5%

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

5. Taylor expanded in f around 0 95.1%

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

   1. associate-*r/ 95.1%

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

   2. mul-1-neg 95.1%

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

   3. unsub-neg 95.1%

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

7. Simplified 95.1%

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

   1. log1p-expm1-u 95.1%

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

   2. expm1-undefine 95.1%

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

   3. diff-log 95.1%

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

   4. add-exp-log 95.1%

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

9. Applied egg-rr 95.1%

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

10. Final simplification 95.1%

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

Alternative 8: 95.6% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.9%

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

3. Simplified 98.5%

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

5. Taylor expanded in f around inf 5.5%

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

   1. associate-*r/ 5.5%

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

7. Simplified 98.7%

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

8. Taylor expanded in f around 0 95.1%

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

   1. associate-/l/ 95.1%

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

10. Simplified 95.1%

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

Alternative 9: 95.5% 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 5.9%

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

3. Simplified 98.5%

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

5. Taylor expanded in f around 0 95.0%

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

6. Final simplification 95.0%

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

Reproduce

herbie shell --seed 2024148 
(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)))))))))