VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 99.1%

Time: 32.0s

Alternatives: 4

Speedup: 4.7×

• 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: 6.8% 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

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

FPCore

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

C

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

Python

def code(f):
	return -4.0 * (math.log(((1.0 / math.expm1((0.5 * (math.pi * f)))) - (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(0.5 * Float64(pi * f)))) - 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[(0.5 * N[(Pi * f), $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 6.7%

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

3. Simplified 98.6%

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

5. Taylor expanded in f around inf 5.9%

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

   1. expm1-define 6.0%

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

   2. *-commutative 6.0%

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

   3. expm1-define 98.8%

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

   4. associate-*r* 98.8%

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

   5. *-commutative 98.8%

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

   6. *-commutative 98.8%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 2: 98.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;\log \left(\frac{f \cdot \left(0.5 - \pi \cdot \left(f \cdot -0.041666666666666664\right)\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{1}{\mathsf{expm1}\left(f \cdot \left(0.5 \cdot \pi\right)\right)}\right) \cdot \frac{-4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{\pi} \cdot 0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 225.0)
   (*
    (log
     (+
      (/
       (+ (* f (- 0.5 (* PI (* f -0.041666666666666664)))) (* 2.0 (/ 1.0 PI)))
       f)
      (/ 1.0 (expm1 (* f (* 0.5 PI))))))
    (/ -4.0 PI))
   (* (/ -4.0 PI) 0.0)))

C

double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = log(((((f * (0.5 - (((double) M_PI) * (f * -0.041666666666666664)))) + (2.0 * (1.0 / ((double) M_PI)))) / f) + (1.0 / expm1((f * (0.5 * ((double) M_PI))))))) * (-4.0 / ((double) M_PI));
	} else {
		tmp = (-4.0 / ((double) M_PI)) * 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = Math.log(((((f * (0.5 - (Math.PI * (f * -0.041666666666666664)))) + (2.0 * (1.0 / Math.PI))) / f) + (1.0 / Math.expm1((f * (0.5 * Math.PI)))))) * (-4.0 / Math.PI);
	} else {
		tmp = (-4.0 / Math.PI) * 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 225.0:
		tmp = math.log(((((f * (0.5 - (math.pi * (f * -0.041666666666666664)))) + (2.0 * (1.0 / math.pi))) / f) + (1.0 / math.expm1((f * (0.5 * math.pi)))))) * (-4.0 / math.pi)
	else:
		tmp = (-4.0 / math.pi) * 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 225.0)
		tmp = Float64(log(Float64(Float64(Float64(Float64(f * Float64(0.5 - Float64(pi * Float64(f * -0.041666666666666664)))) + Float64(2.0 * Float64(1.0 / pi))) / f) + Float64(1.0 / expm1(Float64(f * Float64(0.5 * pi)))))) * Float64(-4.0 / pi));
	else
		tmp = Float64(Float64(-4.0 / pi) * 0.0);
	end
	return tmp
end

Wolfram

code[f_] := If[LessEqual[f, 225.0], N[(N[Log[N[(N[(N[(N[(f * N[(0.5 - N[(Pi * N[(f * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision] + N[(1.0 / N[(Exp[N[(f * N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 / Pi), $MachinePrecision] * 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < 225

   1. Initial program 6.8%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in f around 0 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. *-commutative 98.4%

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

   7. Applied egg-rr 98.4%

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

      1. distribute-lft-out 98.4%

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

      2. *-commutative 98.4%

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

      3. distribute-rgt-out 98.4%

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

      4. metadata-eval 98.4%

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

      5. *-commutative 98.4%

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

      6. associate-*l* 98.4%

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

      7. *-commutative 98.4%

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

   9. Simplified 98.4%

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

   if 225 < f

   1. Initial program 0.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 3.1%

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

      1. rem-exp-log 3.1%

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

      2. flip-+ 0.0%

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

      3. log-div 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

         \[\leadsto \left(\log \left({\left(\frac{-1}{\mathsf{expm1}\left(\left(\pi \cdot -0.5\right) \cdot f\right)}\right)}^{2} - {\left(\frac{-1}{\mathsf{expm1}\left(\left(\pi \cdot -0.5\right) \cdot f\right)}\right)}^{2}\right) - \log \color{blue}{0}\right) \cdot \frac{-4}{\pi} \]

      2. +-inverses 0.0%

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

      3. +-inverses 100.0%

         \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;\log \left(\frac{f \cdot \left(0.5 - \pi \cdot \left(f \cdot -0.041666666666666664\right)\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{1}{\mathsf{expm1}\left(f \cdot \left(0.5 \cdot \pi\right)\right)}\right) \cdot \frac{-4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{\pi} \cdot 0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.25:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{\pi} \cdot 0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.25) (* -4.0 (/ (log (/ (/ 4.0 PI) f)) PI)) (* (/ -4.0 PI) 0.0)))

C

double code(double f) {
	double tmp;
	if (f <= 1.25) {
		tmp = -4.0 * (log(((4.0 / ((double) M_PI)) / f)) / ((double) M_PI));
	} else {
		tmp = (-4.0 / ((double) M_PI)) * 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.25) {
		tmp = -4.0 * (Math.log(((4.0 / Math.PI) / f)) / Math.PI);
	} else {
		tmp = (-4.0 / Math.PI) * 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.25:
		tmp = -4.0 * (math.log(((4.0 / math.pi) / f)) / math.pi)
	else:
		tmp = (-4.0 / math.pi) * 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.25)
		tmp = Float64(-4.0 * Float64(log(Float64(Float64(4.0 / pi) / f)) / pi));
	else
		tmp = Float64(Float64(-4.0 / pi) * 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.25)
		tmp = -4.0 * (log(((4.0 / pi) / f)) / pi);
	else
		tmp = (-4.0 / pi) * 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.25], N[(-4.0 * N[(N[Log[N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 / Pi), $MachinePrecision] * 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < 1.25

   1. Initial program 6.9%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in f around 0 98.9%

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

      1. mul-1-neg 98.9%

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

      2. unsub-neg 98.9%

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

   7. Simplified 98.9%

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

      1. *-un-lft-identity 98.9%

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

      2. diff-log 98.9%

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

   9. Applied egg-rr 98.9%

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

       1. *-lft-identity 98.9%

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

   11. Simplified 98.9%

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

   if 1.25 < f

   1. Initial program 1.1%

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

   3. Simplified 72.5%

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

   6. Applied egg-rr 4.3%

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

      1. rem-exp-log 4.3%

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

      2. flip-+ 0.0%

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

      3. log-div 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

         \[\leadsto \left(\log \left({\left(\frac{-1}{\mathsf{expm1}\left(\left(\pi \cdot -0.5\right) \cdot f\right)}\right)}^{2} - {\left(\frac{-1}{\mathsf{expm1}\left(\left(\pi \cdot -0.5\right) \cdot f\right)}\right)}^{2}\right) - \log \color{blue}{0}\right) \cdot \frac{-4}{\pi} \]

      2. +-inverses 0.0%

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

      3. +-inverses 72.5%

         \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]

   10. Simplified 72.5%

       \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.25:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{\pi} \cdot 0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 5.3% accurate, 106.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

3. Simplified 98.6%

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

6. Applied egg-rr 93.8%

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

   1. rem-exp-log 94.8%

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

   2. flip-+ 0.0%

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

   3. log-div 0.0%

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

8. Applied egg-rr 0.0%

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

   1. +-inverses 0.0%

      \[\leadsto \left(\log \left({\left(\frac{-1}{\mathsf{expm1}\left(\left(\pi \cdot -0.5\right) \cdot f\right)}\right)}^{2} - {\left(\frac{-1}{\mathsf{expm1}\left(\left(\pi \cdot -0.5\right) \cdot f\right)}\right)}^{2}\right) - \log \color{blue}{0}\right) \cdot \frac{-4}{\pi} \]

   2. +-inverses 0.0%

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

   3. +-inverses 5.0%

      \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]

10. Simplified 5.0%

    \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]

11. Final simplification 5.0%

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

Reproduce

herbie shell --seed 2024115 
(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)))))))))