VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 98.9%

Time: 17.1s

Alternatives: 7

Speedup: 532.0×

• 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: 6.9% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.4%

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

3. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 2: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 225.0)
   (-
    (* -4.0 (/ (- (log (/ 4.0 PI)) (log f)) PI))
    (* (pow f 2.0) (* PI 0.08333333333333333)))
   0.0))

C

double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = (-4.0 * ((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI))) - (pow(f, 2.0) * (((double) M_PI) * 0.08333333333333333));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = (-4.0 * ((Math.log((4.0 / Math.PI)) - Math.log(f)) / Math.PI)) - (Math.pow(f, 2.0) * (Math.PI * 0.08333333333333333));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 225.0:
		tmp = (-4.0 * ((math.log((4.0 / math.pi)) - math.log(f)) / math.pi)) - (math.pow(f, 2.0) * (math.pi * 0.08333333333333333))
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 225.0)
		tmp = Float64(Float64(-4.0 * Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi)) - Float64((f ^ 2.0) * Float64(pi * 0.08333333333333333)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 225.0)
		tmp = (-4.0 * ((log((4.0 / pi)) - log(f)) / pi)) - ((f ^ 2.0) * (pi * 0.08333333333333333));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 225.0], N[(N[(-4.0 * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] - N[(N[Power[f, 2.0], $MachinePrecision] * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 225

   1. Initial program 5.6%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in f around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unsub-neg 99.1%

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

      3. mul-1-neg 99.1%

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

      4. unsub-neg 99.1%

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

      5. distribute-rgt-out 99.1%

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

      6. metadata-eval 99.1%

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

   7. Simplified 99.1%

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

   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%

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

      1. count-2 3.1%

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

      2. associate-*r/ 3.1%

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

      3. metadata-eval 3.1%

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

      4. *-commutative 3.1%

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

      5. associate-*r* 3.1%

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

      6. *-commutative 3.1%

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

      7. associate-*l* 3.1%

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

   8. Simplified 3.1%

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

   9. Taylor expanded in f around 0 3.2%

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

   10. Taylor expanded in f around inf 100.0%

       \[\leadsto \log \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 226:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{f \cdot \left(0.5 + f \cdot \left(\pi \cdot 0.041666666666666664\right)\right) + \frac{2}{\pi}}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 226.0)
   (*
    (/ -4.0 PI)
    (log
     (+
      (/ 1.0 (expm1 (* f (* PI 0.5))))
      (/ (+ (* f (+ 0.5 (* f (* PI 0.041666666666666664)))) (/ 2.0 PI)) f))))
   0.0))

C

double code(double f) {
	double tmp;
	if (f <= 226.0) {
		tmp = (-4.0 / ((double) M_PI)) * log(((1.0 / expm1((f * (((double) M_PI) * 0.5)))) + (((f * (0.5 + (f * (((double) M_PI) * 0.041666666666666664)))) + (2.0 / ((double) M_PI))) / f)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 226.0) {
		tmp = (-4.0 / Math.PI) * Math.log(((1.0 / Math.expm1((f * (Math.PI * 0.5)))) + (((f * (0.5 + (f * (Math.PI * 0.041666666666666664)))) + (2.0 / Math.PI)) / f)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 226.0:
		tmp = (-4.0 / math.pi) * math.log(((1.0 / math.expm1((f * (math.pi * 0.5)))) + (((f * (0.5 + (f * (math.pi * 0.041666666666666664)))) + (2.0 / math.pi)) / f)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 226.0)
		tmp = Float64(Float64(-4.0 / pi) * log(Float64(Float64(1.0 / expm1(Float64(f * Float64(pi * 0.5)))) + Float64(Float64(Float64(f * Float64(0.5 + Float64(f * Float64(pi * 0.041666666666666664)))) + Float64(2.0 / pi)) / f))));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[f_] := If[LessEqual[f, 226.0], N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(N[(1.0 / N[(Exp[N[(f * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(f * N[(0.5 + N[(f * N[(Pi * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Pi), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 226

   1. Initial program 5.6%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in f around 0 99.0%

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

      1. pow1 99.0%

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

      2. mul-1-neg 99.0%

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

      3. distribute-rgt-out 99.0%

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

      4. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

      \[\leadsto \log \left(\frac{f \cdot \left(0.5 + \color{blue}{{\left(-f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)}^{1}}\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi} \]
   8. Step-by-step derivation

      1. unpow1 99.0%

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

      2. distribute-rgt-neg-in 99.0%

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

      3. distribute-rgt-neg-in 99.0%

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

      4. metadata-eval 99.0%

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

   9. Simplified 99.0%

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

       1. un-div-inv 99.0%

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

   11. Applied egg-rr 99.0%

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

   if 226 < 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%

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

      1. count-2 3.1%

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

      2. associate-*r/ 3.1%

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

      3. metadata-eval 3.1%

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

      4. *-commutative 3.1%

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

      5. associate-*r* 3.1%

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

      6. *-commutative 3.1%

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

      7. associate-*l* 3.1%

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

   8. Simplified 3.1%

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

   9. Taylor expanded in f around 0 3.2%

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

   10. Taylor expanded in f around inf 100.0%

       \[\leadsto \log \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 226:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{f \cdot \left(0.5 + f \cdot \left(\pi \cdot 0.041666666666666664\right)\right) + \frac{2}{\pi}}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.3:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.3) (* -4.0 (/ (- (log (/ 4.0 PI)) (log f)) PI)) 0.0))

C

double code(double f) {
	double tmp;
	if (f <= 1.3) {
		tmp = -4.0 * ((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.3) {
		tmp = -4.0 * ((Math.log((4.0 / Math.PI)) - Math.log(f)) / Math.PI);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.3:
		tmp = -4.0 * ((math.log((4.0 / math.pi)) - math.log(f)) / math.pi)
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.3)
		tmp = Float64(-4.0 * Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.3)
		tmp = -4.0 * ((log((4.0 / pi)) - log(f)) / pi);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.3], N[(-4.0 * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 1.30000000000000004

   1. Initial program 5.6%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in f around 0 98.8%

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

      1. mul-1-neg 98.8%

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

      2. unsub-neg 98.8%

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

   7. Simplified 98.8%

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

   if 1.30000000000000004 < 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%

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

      1. count-2 3.1%

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

      2. associate-*r/ 3.1%

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

      3. metadata-eval 3.1%

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

      4. *-commutative 3.1%

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

      5. associate-*r* 3.1%

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

      6. *-commutative 3.1%

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

      7. associate-*l* 3.1%

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

   8. Simplified 3.1%

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

   9. Taylor expanded in f around 0 3.2%

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

   10. Taylor expanded in f around inf 100.0%

       \[\leadsto \log \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.3:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.3:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.3) (* -4.0 (/ (log (/ 4.0 (* PI f))) PI)) 0.0))

C

double code(double f) {
	double tmp;
	if (f <= 1.3) {
		tmp = -4.0 * (log((4.0 / (((double) M_PI) * f))) / ((double) M_PI));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.3) {
		tmp = -4.0 * (Math.log((4.0 / (Math.PI * f))) / Math.PI);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.3:
		tmp = -4.0 * (math.log((4.0 / (math.pi * f))) / math.pi)
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.3)
		tmp = Float64(-4.0 * Float64(log(Float64(4.0 / Float64(pi * f))) / pi));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.3)
		tmp = -4.0 * (log((4.0 / (pi * f))) / pi);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.3], N[(-4.0 * N[(N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 1.30000000000000004

   1. Initial program 5.6%

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

   3. Simplified 99.4%

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

   6. Applied egg-rr 0.0%

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

      1. sub-neg 0.0%

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

      2. metadata-eval 0.0%

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

      3. +-commutative 0.0%

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

      4. log1p-undefine 0.0%

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

      5. rem-exp-log 98.1%

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

      6. associate-+r+ 98.1%

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

      7. metadata-eval 98.1%

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

      8. mul0-lft 98.1%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in f around 0 98.8%

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

   if 1.30000000000000004 < 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%

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

      1. count-2 3.1%

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

      2. associate-*r/ 3.1%

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

      3. metadata-eval 3.1%

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

      4. *-commutative 3.1%

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

      5. associate-*r* 3.1%

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

      6. *-commutative 3.1%

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

      7. associate-*l* 3.1%

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

   8. Simplified 3.1%

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

   9. Taylor expanded in f around 0 3.2%

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

   10. Taylor expanded in f around inf 100.0%

       \[\leadsto \log \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.3:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 18.4% accurate, 88.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;-64\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (if (<= f 225.0) -64.0 0.0))

C

double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = -64.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 225.0d0) then
        tmp = -64.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = -64.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 225.0:
		tmp = -64.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 225.0)
		tmp = -64.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 225.0)
		tmp = -64.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 225.0], -64.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 225

   1. Initial program 5.6%

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

   3. Simplified 99.4%

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

   6. Applied egg-rr 0.0%

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

      1. sub-neg 0.0%

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

      2. metadata-eval 0.0%

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

      3. +-commutative 0.0%

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

      4. log1p-undefine 0.0%

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

      5. rem-exp-log 98.1%

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

      6. associate-+r+ 98.1%

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

      7. metadata-eval 98.1%

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

      8. mul0-lft 98.1%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in f around inf 1.9%

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

   11. Simplified 17.2%

       \[\leadsto \color{blue}{-64} \]

   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%

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

      1. count-2 3.1%

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

      2. associate-*r/ 3.1%

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

      3. metadata-eval 3.1%

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

      4. *-commutative 3.1%

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

      5. associate-*r* 3.1%

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

      6. *-commutative 3.1%

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

      7. associate-*l* 3.1%

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

   8. Simplified 3.1%

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

   9. Taylor expanded in f around 0 3.2%

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

   10. Taylor expanded in f around inf 100.0%

       \[\leadsto \log \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;-64\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 16.8% accurate, 532.0× speedup

Math

\[\begin{array}{l} \\ -64 \end{array} \]

FPCore

(FPCore (f) :precision binary64 -64.0)

C

double code(double f) {
	return -64.0;
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = -64.0d0
end function

Java

public static double code(double f) {
	return -64.0;
}

Python

def code(f):
	return -64.0

Julia

function code(f)
	return -64.0
end

MATLAB

function tmp = code(f)
	tmp = -64.0;
end

Wolfram

code[f_] := -64.0

Derivation

1. Initial program 5.4%

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

3. Simplified 99.4%

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

6. Applied egg-rr 0.1%

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

   1. sub-neg 0.1%

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

   2. metadata-eval 0.1%

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

   3. +-commutative 0.1%

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

   4. log1p-undefine 0.1%

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

   5. rem-exp-log 95.9%

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

   6. associate-+r+ 95.9%

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

   7. metadata-eval 95.9%

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

   8. mul0-lft 95.9%

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

8. Simplified 96.0%

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

9. Taylor expanded in f around inf 1.9%

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

11. Simplified 16.9%

    \[\leadsto \color{blue}{-64} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024154 
(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)))))))))