VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.1%

Time: 20.9s

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: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% 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 5.7%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around inf 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 99.0%

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

   1. log1p-expm1-u 99.0%

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

   2. expm1-undefine 99.0%

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

   3. add-exp-log 99.0%

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

9. Applied egg-rr 99.0%

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

    1. sub-neg 99.0%

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

    2. metadata-eval 99.0%

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

    3. distribute-neg-frac 99.0%

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

    4. associate-*r* 99.0%

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

    5. *-commutative 99.0%

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

    6. *-commutative 99.0%

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

    7. expm1-define 4.5%

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

    8. sub-neg 4.5%

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

    9. metadata-eval 4.5%

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

    10. associate-+l- 4.5%

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

11. Simplified 99.0%

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

12. Final simplification 99.0%

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

Alternative 2: 99.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.7%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around inf 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 99.0%

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

8. Final simplification 99.0%

   \[\leadsto \frac{-4 \cdot \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} + \frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(f \cdot \pi\right)\right)}\right)}{\pi} \]
9. 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 5.7%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 4: 96.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.7%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around inf 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 99.0%

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

8. Taylor expanded in f around 0 96.8%

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

9. Final simplification 96.8%

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

Alternative 5: 96.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.7%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around 0 96.6%

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

6. Final simplification 96.6%

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

Alternative 6: 96.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.7%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around 0 96.2%

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

   1. associate-*r/ 96.2%

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

   2. mul-1-neg 96.2%

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

   3. unsub-neg 96.2%

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

7. Simplified 96.2%

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

Alternative 7: 96.1% 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 5.7%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around 0 96.2%

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

   1. associate-*r/ 96.2%

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

   2. mul-1-neg 96.2%

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

   3. unsub-neg 96.2%

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

7. Simplified 96.2%

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

   1. *-un-lft-identity 96.2%

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

   2. diff-log 96.2%

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

9. Applied egg-rr 96.2%

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

    1. *-lft-identity 96.2%

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

11. Simplified 96.2%

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

Alternative 8: 95.9% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.7%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around 0 96.2%

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

   1. associate-*r/ 96.2%

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

   2. mul-1-neg 96.2%

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

   3. unsub-neg 96.2%

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

7. Simplified 96.2%

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

   1. *-commutative 96.2%

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

   2. associate-*r/ 96.1%

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

   3. *-commutative 96.1%

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

   4. diff-log 96.0%

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

9. Applied egg-rr 96.0%

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

Alternative 9: 0.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \frac{-4}{\frac{\pi}{\log 0}} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ -4.0 (/ PI (log 0.0))))

C

double code(double f) {
	return -4.0 / (((double) M_PI) / log(0.0));
}

Java

public static double code(double f) {
	return -4.0 / (Math.PI / Math.log(0.0));
}

Python

def code(f):
	return -4.0 / (math.pi / math.log(0.0))

Julia

function code(f)
	return Float64(-4.0 / Float64(pi / log(0.0)))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 / (pi / log(0.0));
end

Wolfram

code[f_] := N[(-4.0 / N[(Pi / N[Log[0.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.7%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around inf 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 99.0%

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

   1. log1p-expm1-u 99.0%

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

   2. expm1-undefine 99.0%

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

   3. add-exp-log 99.0%

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

9. Applied egg-rr 99.0%

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

    1. sub-neg 99.0%

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

    2. metadata-eval 99.0%

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

    3. distribute-neg-frac 99.0%

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

    4. associate-*r* 99.0%

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

    5. *-commutative 99.0%

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

    6. *-commutative 99.0%

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

    7. expm1-define 4.5%

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

    8. sub-neg 4.5%

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

    9. metadata-eval 4.5%

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

    10. associate-+l- 4.5%

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

11. Simplified 99.0%

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

    1. clear-num 98.8%

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

    2. inv-pow 98.8%

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

13. Applied egg-rr 0.7%

    \[\leadsto \color{blue}{{\left(-0.25 \cdot \frac{\pi}{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)}\right) + -1\right)}\right)}^{-1}} \]
14. Step-by-step derivation

    1. unpow-1 0.7%

       \[\leadsto \color{blue}{\frac{1}{-0.25 \cdot \frac{\pi}{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)}\right) + -1\right)}}} \]

    2. associate-/r* 0.7%

       \[\leadsto \color{blue}{\frac{\frac{1}{-0.25}}{\frac{\pi}{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)}\right) + -1\right)}}} \]

    3. metadata-eval 0.7%

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

    4. log1p-undefine 0.7%

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

15. Simplified 0.7%

    \[\leadsto \color{blue}{\frac{-4}{\frac{\pi}{\log 0}}} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024108 
(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)))))))))