VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.1%

Time: 31.1s

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: 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 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 99.0%

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

5. Taylor expanded in f around inf 5.6%

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

7. Simplified 99.2%

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

Alternative 2: 98.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\\ \mathbf{if}\;f \leq 2.4:\\ \;\;\;\;-4 \cdot \frac{\log \left(t\_0 + \frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + \pi \cdot \left(f \cdot -0.041666666666666664\right)\right)}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\log t\_0 \cdot \frac{-4}{\pi}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ -1.0 (expm1 (* PI (* f -0.5))))))
   (if (<= f 2.4)
     (*
      -4.0
      (/
       (log
        (+
         t_0
         (/
          (-
           (* 2.0 (/ 1.0 PI))
           (* f (+ 0.5 (* PI (* f -0.041666666666666664)))))
          f)))
       PI))
     (* (log t_0) (/ -4.0 PI)))))

C

double code(double f) {
	double t_0 = -1.0 / expm1((((double) M_PI) * (f * -0.5)));
	double tmp;
	if (f <= 2.4) {
		tmp = -4.0 * (log((t_0 + (((2.0 * (1.0 / ((double) M_PI))) - (f * (0.5 + (((double) M_PI) * (f * -0.041666666666666664))))) / f))) / ((double) M_PI));
	} else {
		tmp = log(t_0) * (-4.0 / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = -1.0 / Math.expm1((Math.PI * (f * -0.5)));
	double tmp;
	if (f <= 2.4) {
		tmp = -4.0 * (Math.log((t_0 + (((2.0 * (1.0 / Math.PI)) - (f * (0.5 + (Math.PI * (f * -0.041666666666666664))))) / f))) / Math.PI);
	} else {
		tmp = Math.log(t_0) * (-4.0 / Math.PI);
	}
	return tmp;
}

Python

def code(f):
	t_0 = -1.0 / math.expm1((math.pi * (f * -0.5)))
	tmp = 0
	if f <= 2.4:
		tmp = -4.0 * (math.log((t_0 + (((2.0 * (1.0 / math.pi)) - (f * (0.5 + (math.pi * (f * -0.041666666666666664))))) / f))) / math.pi)
	else:
		tmp = math.log(t_0) * (-4.0 / math.pi)
	return tmp

Julia

function code(f)
	t_0 = Float64(-1.0 / expm1(Float64(pi * Float64(f * -0.5))))
	tmp = 0.0
	if (f <= 2.4)
		tmp = Float64(-4.0 * Float64(log(Float64(t_0 + Float64(Float64(Float64(2.0 * Float64(1.0 / pi)) - Float64(f * Float64(0.5 + Float64(pi * Float64(f * -0.041666666666666664))))) / f))) / pi));
	else
		tmp = Float64(log(t_0) * Float64(-4.0 / pi));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[(-1.0 / N[(Exp[N[(Pi * N[(f * -0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[f, 2.4], N[(-4.0 * N[(N[Log[N[(t$95$0 + 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] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[Log[t$95$0], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if f < 2.39999999999999991

   1. Initial program 5.8%

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

   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 2.9%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in f around 0 99.3%

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

      1. distribute-lft-in 99.3%

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

      2. *-commutative 99.3%

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

      3. *-commutative 99.3%

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

   10. Applied egg-rr 99.3%

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

       1. distribute-lft-out 99.3%

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

       2. *-commutative 99.3%

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

       3. *-commutative 99.3%

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

       4. *-commutative 99.3%

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

       5. distribute-rgt-out 99.3%

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

       6. metadata-eval 99.3%

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

       7. associate-*l* 99.3%

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

   12. Simplified 99.3%

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

   if 2.39999999999999991 < f

   1. Initial program 3.0%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in f around 0 4.0%

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

      1. *-commutative 4.0%

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

   7. Simplified 4.0%

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

   8. Taylor expanded in f around inf 89.9%

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

      1. distribute-neg-frac 89.9%

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

      2. metadata-eval 89.9%

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

      3. expm1-define 89.9%

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

      4. associate-*r* 89.9%

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

      5. *-commutative 89.9%

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

      6. *-commutative 89.9%

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

   10. Simplified 89.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 2.4:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} + \frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + \pi \cdot \left(f \cdot -0.041666666666666664\right)\right)}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right) \cdot \frac{-4}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if f < 1

   1. Initial program 5.8%

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

   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 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unsub-neg 99.1%

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

   7. Simplified 99.1%

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

      1. *-un-lft-identity 99.1%

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

      2. diff-log 99.1%

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

   9. Applied egg-rr 99.1%

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

       1. *-lft-identity 99.1%

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

       2. associate-/l/ 99.1%

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

       3. *-commutative 99.1%

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

   11. Simplified 99.1%

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

   12. Taylor expanded in f around inf 99.1%

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

   if 1 < f

   1. Initial program 3.0%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in f around 0 4.0%

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

      1. *-commutative 4.0%

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

   7. Simplified 4.0%

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

   8. Taylor expanded in f around inf 89.9%

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

      1. distribute-neg-frac 89.9%

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

      2. metadata-eval 89.9%

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

      3. expm1-define 89.9%

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

      4. associate-*r* 89.9%

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

      5. *-commutative 89.9%

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

      6. *-commutative 89.9%

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

   10. Simplified 89.9%

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

Alternative 4: 95.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(-4.0 * N[(N[(N[Log[N[(1.0 / f), $MachinePrecision]], $MachinePrecision] + N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 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 99.0%

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

5. Taylor expanded in f around 0 96.1%

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

   1. mul-1-neg 96.1%

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

   2. unsub-neg 96.1%

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

7. Simplified 96.1%

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

   1. *-un-lft-identity 96.1%

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

   2. diff-log 96.1%

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

9. Applied egg-rr 96.1%

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

    1. *-lft-identity 96.1%

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

    2. associate-/l/ 96.1%

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

    3. *-commutative 96.1%

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

11. Simplified 96.1%

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

12. Taylor expanded in f around inf 96.1%

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

Alternative 5: 96.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\\ t_1 := 2 \cdot \frac{1}{\pi}\\ \frac{-4}{\pi} \cdot \log \left(\frac{t\_1 - f \cdot \left(0.5 + t\_0\right)}{f} + \frac{t\_1 + f \cdot \left(0.5 - t\_0\right)}{f}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* f (+ (* PI -0.125) (* PI 0.08333333333333333))))
        (t_1 (* 2.0 (/ 1.0 PI))))
   (*
    (/ -4.0 PI)
    (log
     (+ (/ (- t_1 (* f (+ 0.5 t_0))) f) (/ (+ t_1 (* f (- 0.5 t_0))) f))))))

C

double code(double f) {
	double t_0 = f * ((((double) M_PI) * -0.125) + (((double) M_PI) * 0.08333333333333333));
	double t_1 = 2.0 * (1.0 / ((double) M_PI));
	return (-4.0 / ((double) M_PI)) * log((((t_1 - (f * (0.5 + t_0))) / f) + ((t_1 + (f * (0.5 - t_0))) / f)));
}

Java

public static double code(double f) {
	double t_0 = f * ((Math.PI * -0.125) + (Math.PI * 0.08333333333333333));
	double t_1 = 2.0 * (1.0 / Math.PI);
	return (-4.0 / Math.PI) * Math.log((((t_1 - (f * (0.5 + t_0))) / f) + ((t_1 + (f * (0.5 - t_0))) / f)));
}

Python

def code(f):
	t_0 = f * ((math.pi * -0.125) + (math.pi * 0.08333333333333333))
	t_1 = 2.0 * (1.0 / math.pi)
	return (-4.0 / math.pi) * math.log((((t_1 - (f * (0.5 + t_0))) / f) + ((t_1 + (f * (0.5 - t_0))) / f)))

Julia

function code(f)
	t_0 = Float64(f * Float64(Float64(pi * -0.125) + Float64(pi * 0.08333333333333333)))
	t_1 = Float64(2.0 * Float64(1.0 / pi))
	return Float64(Float64(-4.0 / pi) * log(Float64(Float64(Float64(t_1 - Float64(f * Float64(0.5 + t_0))) / f) + Float64(Float64(t_1 + Float64(f * Float64(0.5 - t_0))) / f))))
end

MATLAB

function tmp = code(f)
	t_0 = f * ((pi * -0.125) + (pi * 0.08333333333333333));
	t_1 = 2.0 * (1.0 / pi);
	tmp = (-4.0 / pi) * log((((t_1 - (f * (0.5 + t_0))) / f) + ((t_1 + (f * (0.5 - t_0))) / f)));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(f * N[(N[(Pi * -0.125), $MachinePrecision] + N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(N[(N[(t$95$1 - N[(f * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision] + N[(N[(t$95$1 + N[(f * N[(0.5 - t$95$0), $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 99.0%

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

5. Taylor expanded in f around 0 96.1%

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

6. Taylor expanded in f around 0 96.1%

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

7. Final simplification 96.1%

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

Alternative 6: 95.5% accurate, 4.9× speedup

Math

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

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(4.0 / Float64(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[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 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 99.0%

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

5. Taylor expanded in f around 0 96.1%

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

   1. mul-1-neg 96.1%

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

   2. unsub-neg 96.1%

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

7. Simplified 96.1%

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

   1. *-un-lft-identity 96.1%

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

   2. diff-log 96.1%

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

9. Applied egg-rr 96.1%

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

    1. *-lft-identity 96.1%

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

    2. associate-/l/ 96.1%

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

    3. *-commutative 96.1%

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

11. Simplified 96.1%

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

Reproduce

herbie shell --seed 2024091 
(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)))))))))