VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.0% → 98.9%

Time: 37.1s

Alternatives: 6

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: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in f around inf 6.1%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 96.4% accurate, 2.3× speedup

Math

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

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

Derivation

1. Initial program 7.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in f around inf 6.1%

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

7. Simplified 99.5%

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

8. Taylor expanded in f around 0 97.7%

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

9. Final simplification 97.7%

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

Alternative 3: 96.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} + \frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + \pi \cdot \left(f \cdot -0.041666666666666664\right)\right)}{f}\right) \cdot \frac{-4}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (+
    (/ -1.0 (expm1 (* PI (* f -0.5))))
    (/
     (- (* 2.0 (/ 1.0 PI)) (* f (+ 0.5 (* PI (* f -0.041666666666666664)))))
     f)))
  (/ -4.0 PI)))

C

double code(double f) {
	return log(((-1.0 / expm1((((double) M_PI) * (f * -0.5)))) + (((2.0 * (1.0 / ((double) M_PI))) - (f * (0.5 + (((double) M_PI) * (f * -0.041666666666666664))))) / f))) * (-4.0 / ((double) M_PI));
}

Java

public static double code(double f) {
	return Math.log(((-1.0 / Math.expm1((Math.PI * (f * -0.5)))) + (((2.0 * (1.0 / Math.PI)) - (f * (0.5 + (Math.PI * (f * -0.041666666666666664))))) / f))) * (-4.0 / Math.PI);
}

Python

def code(f):
	return math.log(((-1.0 / math.expm1((math.pi * (f * -0.5)))) + (((2.0 * (1.0 / math.pi)) - (f * (0.5 + (math.pi * (f * -0.041666666666666664))))) / f))) * (-4.0 / math.pi)

Julia

function code(f)
	return Float64(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(pi * Float64(f * -0.041666666666666664))))) / f))) * 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[(N[(N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] - N[(f * N[(0.5 + N[(Pi * N[(f * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in f around 0 97.5%

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

   1. distribute-lft-in 97.5%

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

   2. *-commutative 97.5%

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

   3. *-commutative 97.5%

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

7. Applied egg-rr 97.5%

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

   1. associate-*r* 97.5%

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

   2. associate-*r* 97.5%

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

   3. distribute-lft-out 97.5%

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

   4. metadata-eval 97.5%

      \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \left(\left(f \cdot \pi\right) \cdot \color{blue}{-0.041666666666666664}\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   5. *-commutative 97.5%

      \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \left(\color{blue}{\left(\pi \cdot f\right)} \cdot -0.041666666666666664\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   6. associate-*l* 97.5%

      \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \color{blue}{\left(\pi \cdot \left(f \cdot -0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

9. Simplified 97.5%

   \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \color{blue}{\left(\pi \cdot \left(f \cdot -0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi} \]

10. Final simplification 97.5%

    \[\leadsto \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} + \frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + \pi \cdot \left(f \cdot -0.041666666666666664\right)\right)}{f}\right) \cdot \frac{-4}{\pi} \]
11. Add Preprocessing

Alternative 4: 96.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \frac{1}{\pi}\\ \frac{-4}{\pi} \cdot \log \left(\frac{t\_0 - f \cdot \left(0.5 + \pi \cdot \left(f \cdot -0.041666666666666664\right)\right)}{f} + \frac{t\_0 + f \cdot \left(0.5 - f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)}{f}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* 2.0 (/ 1.0 PI))))
   (*
    (/ -4.0 PI)
    (log
     (+
      (/ (- t_0 (* f (+ 0.5 (* PI (* f -0.041666666666666664))))) f)
      (/
       (+ t_0 (* f (- 0.5 (* f (+ (* PI -0.125) (* PI 0.08333333333333333))))))
       f))))))

C

double code(double f) {
	double t_0 = 2.0 * (1.0 / ((double) M_PI));
	return (-4.0 / ((double) M_PI)) * log((((t_0 - (f * (0.5 + (((double) M_PI) * (f * -0.041666666666666664))))) / f) + ((t_0 + (f * (0.5 - (f * ((((double) M_PI) * -0.125) + (((double) M_PI) * 0.08333333333333333)))))) / f)));
}

Java

public static double code(double f) {
	double t_0 = 2.0 * (1.0 / Math.PI);
	return (-4.0 / Math.PI) * Math.log((((t_0 - (f * (0.5 + (Math.PI * (f * -0.041666666666666664))))) / f) + ((t_0 + (f * (0.5 - (f * ((Math.PI * -0.125) + (Math.PI * 0.08333333333333333)))))) / f)));
}

Python

def code(f):
	t_0 = 2.0 * (1.0 / math.pi)
	return (-4.0 / math.pi) * math.log((((t_0 - (f * (0.5 + (math.pi * (f * -0.041666666666666664))))) / f) + ((t_0 + (f * (0.5 - (f * ((math.pi * -0.125) + (math.pi * 0.08333333333333333)))))) / f)))

Julia

function code(f)
	t_0 = Float64(2.0 * Float64(1.0 / pi))
	return Float64(Float64(-4.0 / pi) * log(Float64(Float64(Float64(t_0 - Float64(f * Float64(0.5 + Float64(pi * Float64(f * -0.041666666666666664))))) / f) + Float64(Float64(t_0 + Float64(f * Float64(0.5 - Float64(f * Float64(Float64(pi * -0.125) + Float64(pi * 0.08333333333333333)))))) / f))))
end

MATLAB

function tmp = code(f)
	t_0 = 2.0 * (1.0 / pi);
	tmp = (-4.0 / pi) * log((((t_0 - (f * (0.5 + (pi * (f * -0.041666666666666664))))) / f) + ((t_0 + (f * (0.5 - (f * ((pi * -0.125) + (pi * 0.08333333333333333)))))) / f)));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(N[(N[(t$95$0 - N[(f * N[(0.5 + N[(Pi * N[(f * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision] + N[(N[(t$95$0 + 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 7.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in f around 0 97.5%

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

6. Taylor expanded in f around 0 97.5%

   \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \color{blue}{\frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}}\right) \cdot \frac{-4}{\pi} \]
7. Step-by-step derivation

   1. distribute-lft-in 97.5%

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

   2. *-commutative 97.5%

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

   3. *-commutative 97.5%

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

8. Applied egg-rr 97.5%

   \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \color{blue}{\left(f \cdot \left(\pi \cdot -0.125\right) + f \cdot \left(\pi \cdot 0.08333333333333333\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]
9. Step-by-step derivation

   1. associate-*r* 97.5%

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

   2. associate-*r* 97.5%

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

   3. distribute-lft-out 97.5%

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

   4. metadata-eval 97.5%

      \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \left(\left(f \cdot \pi\right) \cdot \color{blue}{-0.041666666666666664}\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   5. *-commutative 97.5%

      \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \left(\color{blue}{\left(\pi \cdot f\right)} \cdot -0.041666666666666664\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   6. associate-*l* 97.5%

      \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \color{blue}{\left(\pi \cdot \left(f \cdot -0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

10. Simplified 97.5%

    \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \color{blue}{\left(\pi \cdot \left(f \cdot -0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

11. Final simplification 97.5%

    \[\leadsto \frac{-4}{\pi} \cdot \log \left(\frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + \pi \cdot \left(f \cdot -0.041666666666666664\right)\right)}{f} + \frac{2 \cdot \frac{1}{\pi} + f \cdot \left(0.5 - f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)}{f}\right) \]
12. Add Preprocessing

Alternative 5: 95.8% accurate, 4.9× speedup

Math

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

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

Derivation

1. Initial program 7.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in f around 0 96.9%

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

   1. mul-1-neg 96.9%

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

   2. unsub-neg 96.9%

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

7. Simplified 96.9%

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

   1. *-un-lft-identity 96.9%

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

   2. diff-log 96.9%

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

9. Applied egg-rr 96.9%

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

    1. *-lft-identity 96.9%

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

11. Simplified 96.9%

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

12. Final simplification 96.9%

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

Alternative 6: 5.0% accurate, 106.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

3. Simplified 99.3%

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

5. Taylor expanded in f around 0 96.9%

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

   1. mul-1-neg 96.9%

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

   2. unsub-neg 96.9%

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

7. Simplified 96.9%

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

   1. *-un-lft-identity 96.9%

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

   2. diff-log 96.9%

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

9. Applied egg-rr 96.9%

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

    1. *-lft-identity 96.9%

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

11. Simplified 96.9%

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

    1. clear-num 96.9%

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

    2. log-div 96.9%

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

    3. metadata-eval 96.9%

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

    4. div-inv 96.9%

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

    5. clear-num 96.9%

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

    6. div-inv 96.9%

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

    7. metadata-eval 96.9%

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

13. Applied egg-rr 96.9%

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

    1. neg-sub0 96.9%

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

    2. *-commutative 96.9%

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

    3. associate-*l* 96.9%

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

15. Simplified 96.9%

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

17. Applied egg-rr 4.2%

    \[\leadsto -4 \cdot \frac{-\log \color{blue}{\left(\frac{\frac{2}{\sqrt{\pi \cdot f}}}{\frac{2}{\sqrt{\pi \cdot f}}}\right)}}{\pi} \]
18. Step-by-step derivation

    1. *-inverses 4.2%

       \[\leadsto -4 \cdot \frac{-\log \color{blue}{1}}{\pi} \]

19. Simplified 4.2%

    \[\leadsto -4 \cdot \frac{-\log \color{blue}{1}}{\pi} \]

20. Final simplification 4.2%

    \[\leadsto -4 \cdot \frac{0}{\pi} \]
21. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(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)))))))))