VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 99.1%

Time: 17.6s

Alternatives: 8

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1307889555.4486573

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

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

FPCore

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

C

double code(double f) {
	return -4.0 * (log(((1.0 / expm1((f * (((double) M_PI) * 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((f * (Math.PI * 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((f * (math.pi * 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(f * Float64(pi * 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[(f * N[(Pi * 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 6.2%

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

3. Simplified 99.0%

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

5. Taylor expanded in f around inf 5.7%

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

   1. expm1-define 5.8%

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

   2. *-commutative 5.8%

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

   3. associate-*l* 5.8%

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

   4. expm1-define 99.1%

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

   5. associate-*r* 99.1%

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

   6. *-commutative 99.1%

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

   7. *-commutative 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 2: 98.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.6:\\ \;\;\;\;-4 \cdot \left(\left(\frac{\log \left(\frac{4}{\pi}\right)}{\pi} + {f}^{2} \cdot \left(\pi \cdot 0.020833333333333332\right)\right) - \frac{\log f}{\pi}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.6)
   (*
    -4.0
    (-
     (+ (/ (log (/ 4.0 PI)) PI) (* (pow f 2.0) (* PI 0.020833333333333332)))
     (/ (log f) PI)))
   (* -4.0 (/ (log (/ -1.0 (expm1 (* PI (* f -0.5))))) PI))))

C

double code(double f) {
	double tmp;
	if (f <= 1.6) {
		tmp = -4.0 * (((log((4.0 / ((double) M_PI))) / ((double) M_PI)) + (pow(f, 2.0) * (((double) M_PI) * 0.020833333333333332))) - (log(f) / ((double) M_PI)));
	} else {
		tmp = -4.0 * (log((-1.0 / expm1((((double) M_PI) * (f * -0.5))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.6) {
		tmp = -4.0 * (((Math.log((4.0 / Math.PI)) / Math.PI) + (Math.pow(f, 2.0) * (Math.PI * 0.020833333333333332))) - (Math.log(f) / Math.PI));
	} else {
		tmp = -4.0 * (Math.log((-1.0 / Math.expm1((Math.PI * (f * -0.5))))) / Math.PI);
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.6:
		tmp = -4.0 * (((math.log((4.0 / math.pi)) / math.pi) + (math.pow(f, 2.0) * (math.pi * 0.020833333333333332))) - (math.log(f) / math.pi))
	else:
		tmp = -4.0 * (math.log((-1.0 / math.expm1((math.pi * (f * -0.5))))) / math.pi)
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.6)
		tmp = Float64(-4.0 * Float64(Float64(Float64(log(Float64(4.0 / pi)) / pi) + Float64((f ^ 2.0) * Float64(pi * 0.020833333333333332))) - Float64(log(f) / pi)));
	else
		tmp = Float64(-4.0 * Float64(log(Float64(-1.0 / expm1(Float64(pi * Float64(f * -0.5))))) / pi));
	end
	return tmp
end

Wolfram

code[f_] := If[LessEqual[f, 1.6], N[(-4.0 * N[(N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] + N[(N[Power[f, 2.0], $MachinePrecision] * N[(Pi * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Log[f], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[Log[N[(-1.0 / N[(Exp[N[(Pi * N[(f * -0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < 1.6000000000000001

   1. Initial program 6.0%

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

   3. Simplified 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 inf 3.1%

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

      1. expm1-define 3.2%

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

      2. *-commutative 3.2%

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

      3. associate-*l* 3.2%

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

      4. expm1-define 99.5%

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

      5. associate-*r* 99.5%

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

      6. *-commutative 99.5%

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

      7. *-commutative 99.5%

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

   7. Simplified 99.5%

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

      1. add-cbrt-cube 98.9%

         \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}} \]

      2. pow3 98.9%

         \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\sqrt[3]{\color{blue}{{\pi}^{3}}}} \]

   9. Applied egg-rr 98.9%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\color{blue}{\sqrt[3]{{\pi}^{3}}}} \]
   10. Step-by-step derivation

       1. add-log-exp 98.9%

          \[\leadsto -4 \cdot \color{blue}{\log \left(e^{\frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\sqrt[3]{{\pi}^{3}}}}\right)} \]

       2. rem-cbrt-cube 99.5%

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

       3. div-inv 99.4%

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

       4. exp-to-pow 99.4%

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

   11. Applied egg-rr 99.4%

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

   12. Taylor expanded in f around 0 99.0%

       \[\leadsto -4 \cdot \color{blue}{\left(-1 \cdot \frac{\log f}{\pi} + \left(0.25 \cdot \left({f}^{2} \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right)\right)} \]
   13. Step-by-step derivation

       1. +-commutative 99.0%

          \[\leadsto -4 \cdot \color{blue}{\left(\left(0.25 \cdot \left({f}^{2} \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right) + -1 \cdot \frac{\log f}{\pi}\right)} \]

       2. mul-1-neg 99.0%

          \[\leadsto -4 \cdot \left(\left(0.25 \cdot \left({f}^{2} \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right) + \color{blue}{\left(-\frac{\log f}{\pi}\right)}\right) \]

       3. unsub-neg 99.0%

          \[\leadsto -4 \cdot \color{blue}{\left(\left(0.25 \cdot \left({f}^{2} \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right) - \frac{\log f}{\pi}\right)} \]

   14. Simplified 99.0%

       \[\leadsto -4 \cdot \color{blue}{\left(\left(\frac{\log \left(\frac{4}{\pi}\right)}{\pi} + {f}^{2} \cdot \left(0.020833333333333332 \cdot \pi\right)\right) - \frac{\log f}{\pi}\right)} \]

   if 1.6000000000000001 < f

   1. Initial program 11.0%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in f around inf 86.0%

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

      1. expm1-define 86.0%

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

      2. *-commutative 86.0%

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

      3. associate-*l* 86.0%

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

      4. expm1-define 86.0%

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

      5. associate-*r* 86.0%

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

      6. *-commutative 86.0%

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

      7. *-commutative 86.0%

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

   7. Simplified 86.0%

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

   8. Taylor expanded in f around 0 17.6%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\color{blue}{f \cdot \left(0.5 \cdot \pi + f \cdot \left(0.020833333333333332 \cdot \left(f \cdot {\pi}^{3}\right) + 0.125 \cdot {\pi}^{2}\right)\right)}} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi} \]

   9. Taylor expanded in f around inf 77.8%

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

       1. distribute-neg-frac 77.8%

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

       2. metadata-eval 77.8%

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

       3. expm1-define 77.8%

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

       4. *-commutative 77.8%

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

       5. *-commutative 77.8%

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

       6. associate-*r* 77.8%

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

   11. Simplified 77.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.6:\\ \;\;\;\;-4 \cdot \left(\left(\frac{\log \left(\frac{4}{\pi}\right)}{\pi} + {f}^{2} \cdot \left(\pi \cdot 0.020833333333333332\right)\right) - \frac{\log f}{\pi}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 230:\\ \;\;\;\;\log \left(\frac{f \cdot \left(0.5 - f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)}\right) \cdot \frac{-4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 230.0)
   (*
    (log
     (+
      (/
       (+
        (* f (- 0.5 (* f (+ (* PI -0.125) (* PI 0.08333333333333333)))))
        (* 2.0 (/ 1.0 PI)))
       f)
      (/ 1.0 (expm1 (* f (* PI 0.5))))))
    (/ -4.0 PI))
   0.0))

C

double code(double f) {
	double tmp;
	if (f <= 230.0) {
		tmp = log(((((f * (0.5 - (f * ((((double) M_PI) * -0.125) + (((double) M_PI) * 0.08333333333333333))))) + (2.0 * (1.0 / ((double) M_PI)))) / f) + (1.0 / expm1((f * (((double) M_PI) * 0.5)))))) * (-4.0 / ((double) M_PI));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 230.0) {
		tmp = Math.log(((((f * (0.5 - (f * ((Math.PI * -0.125) + (Math.PI * 0.08333333333333333))))) + (2.0 * (1.0 / Math.PI))) / f) + (1.0 / Math.expm1((f * (Math.PI * 0.5)))))) * (-4.0 / Math.PI);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 230.0:
		tmp = math.log(((((f * (0.5 - (f * ((math.pi * -0.125) + (math.pi * 0.08333333333333333))))) + (2.0 * (1.0 / math.pi))) / f) + (1.0 / math.expm1((f * (math.pi * 0.5)))))) * (-4.0 / math.pi)
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 230.0)
		tmp = Float64(log(Float64(Float64(Float64(Float64(f * Float64(0.5 - Float64(f * Float64(Float64(pi * -0.125) + Float64(pi * 0.08333333333333333))))) + Float64(2.0 * Float64(1.0 / pi))) / f) + Float64(1.0 / expm1(Float64(f * Float64(pi * 0.5)))))) * Float64(-4.0 / pi));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[f_] := If[LessEqual[f, 230.0], N[(N[Log[N[(N[(N[(N[(f * N[(0.5 - N[(f * N[(N[(Pi * -0.125), $MachinePrecision] + N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $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], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 230

   1. Initial program 6.3%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in f around 0 98.2%

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

   if 230 < 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 1.6%

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

      1. +-inverses 1.6%

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

   8. Simplified 1.6%

      \[\leadsto \log \color{blue}{0} \cdot \frac{-4}{\pi} \]
   9. Step-by-step derivation

      1. +-inverses 1.6%

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

      2. sub-neg 1.6%

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

      3. add-sqr-sqrt 1.6%

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

      4. sqrt-unprod 1.6%

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

      5. *-commutative 1.6%

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

      6. *-commutative 1.6%

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

      7. frac-times 1.6%

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

      8. metadata-eval 1.6%

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

      9. metadata-eval 1.6%

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

      10. frac-times 1.6%

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

   10. Applied egg-rr 0.0%

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

       1. +-inverses 100.0%

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

   12. Simplified 100.0%

       \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]
   13. Step-by-step derivation

       1. mul0-lft 100.0%

          \[\leadsto \color{blue}{0} \]

   14. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 230:\\ \;\;\;\;\log \left(\frac{f \cdot \left(0.5 - f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)}\right) \cdot \frac{-4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 230:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{{f}^{2} \cdot \left(\left(\pi \cdot -0.08333333333333333 + \pi \cdot 0.125\right) - \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right) + 4 \cdot \frac{1}{\pi}}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 230.0)
   (*
    (/ -4.0 PI)
    (log
     (/
      (+
       (*
        (pow f 2.0)
        (-
         (+ (* PI -0.08333333333333333) (* PI 0.125))
         (+ (* PI -0.125) (* PI 0.08333333333333333))))
       (* 4.0 (/ 1.0 PI)))
      f)))
   0.0))

C

double code(double f) {
	double tmp;
	if (f <= 230.0) {
		tmp = (-4.0 / ((double) M_PI)) * log((((pow(f, 2.0) * (((((double) M_PI) * -0.08333333333333333) + (((double) M_PI) * 0.125)) - ((((double) M_PI) * -0.125) + (((double) M_PI) * 0.08333333333333333)))) + (4.0 * (1.0 / ((double) M_PI)))) / f));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 230.0) {
		tmp = (-4.0 / Math.PI) * Math.log((((Math.pow(f, 2.0) * (((Math.PI * -0.08333333333333333) + (Math.PI * 0.125)) - ((Math.PI * -0.125) + (Math.PI * 0.08333333333333333)))) + (4.0 * (1.0 / Math.PI))) / f));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 230.0:
		tmp = (-4.0 / math.pi) * math.log((((math.pow(f, 2.0) * (((math.pi * -0.08333333333333333) + (math.pi * 0.125)) - ((math.pi * -0.125) + (math.pi * 0.08333333333333333)))) + (4.0 * (1.0 / math.pi))) / f))
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 230.0)
		tmp = Float64(Float64(-4.0 / pi) * log(Float64(Float64(Float64((f ^ 2.0) * Float64(Float64(Float64(pi * -0.08333333333333333) + Float64(pi * 0.125)) - Float64(Float64(pi * -0.125) + Float64(pi * 0.08333333333333333)))) + Float64(4.0 * Float64(1.0 / pi))) / f)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 230.0)
		tmp = (-4.0 / pi) * log(((((f ^ 2.0) * (((pi * -0.08333333333333333) + (pi * 0.125)) - ((pi * -0.125) + (pi * 0.08333333333333333)))) + (4.0 * (1.0 / pi))) / f));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 230.0], N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(N[(N[(N[Power[f, 2.0], $MachinePrecision] * N[(N[(N[(Pi * -0.08333333333333333), $MachinePrecision] + N[(Pi * 0.125), $MachinePrecision]), $MachinePrecision] - N[(N[(Pi * -0.125), $MachinePrecision] + N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 230

   1. Initial program 6.3%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in f around 0 98.2%

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

   if 230 < 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 1.6%

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

      1. +-inverses 1.6%

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

   8. Simplified 1.6%

      \[\leadsto \log \color{blue}{0} \cdot \frac{-4}{\pi} \]
   9. Step-by-step derivation

      1. +-inverses 1.6%

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

      2. sub-neg 1.6%

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

      3. add-sqr-sqrt 1.6%

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

      4. sqrt-unprod 1.6%

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

      5. *-commutative 1.6%

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

      6. *-commutative 1.6%

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

      7. frac-times 1.6%

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

      8. metadata-eval 1.6%

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

      9. metadata-eval 1.6%

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

      10. frac-times 1.6%

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

   10. Applied egg-rr 0.0%

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

       1. +-inverses 100.0%

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

   12. Simplified 100.0%

       \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]
   13. Step-by-step derivation

       1. mul0-lft 100.0%

          \[\leadsto \color{blue}{0} \]

   14. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 230:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{{f}^{2} \cdot \left(\left(\pi \cdot -0.08333333333333333 + \pi \cdot 0.125\right) - \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right) + 4 \cdot \frac{1}{\pi}}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.05:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{f \cdot \pi}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.05)
   (* -4.0 (/ (log (/ 4.0 (* f PI))) PI))
   (* -4.0 (/ (log (/ -1.0 (expm1 (* PI (* f -0.5))))) PI))))

C

double code(double f) {
	double tmp;
	if (f <= 1.05) {
		tmp = -4.0 * (log((4.0 / (f * ((double) M_PI)))) / ((double) M_PI));
	} else {
		tmp = -4.0 * (log((-1.0 / expm1((((double) M_PI) * (f * -0.5))))) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.05) {
		tmp = -4.0 * (Math.log((4.0 / (f * Math.PI))) / Math.PI);
	} else {
		tmp = -4.0 * (Math.log((-1.0 / Math.expm1((Math.PI * (f * -0.5))))) / Math.PI);
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.05:
		tmp = -4.0 * (math.log((4.0 / (f * math.pi))) / math.pi)
	else:
		tmp = -4.0 * (math.log((-1.0 / math.expm1((math.pi * (f * -0.5))))) / math.pi)
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.05)
		tmp = Float64(-4.0 * Float64(log(Float64(4.0 / Float64(f * pi))) / pi));
	else
		tmp = Float64(-4.0 * Float64(log(Float64(-1.0 / expm1(Float64(pi * Float64(f * -0.5))))) / pi));
	end
	return tmp
end

Wolfram

code[f_] := If[LessEqual[f, 1.05], N[(-4.0 * N[(N[Log[N[(4.0 / N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[Log[N[(-1.0 / N[(Exp[N[(Pi * N[(f * -0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < 1.05000000000000004

   1. Initial program 6.0%

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

   3. Simplified 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 inf 3.1%

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

      1. expm1-define 3.2%

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

      2. *-commutative 3.2%

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

      3. associate-*l* 3.2%

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

      4. expm1-define 99.5%

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

      5. associate-*r* 99.5%

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

      6. *-commutative 99.5%

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

      7. *-commutative 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in f around 0 98.6%

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

      1. *-commutative 98.6%

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

   10. Simplified 98.6%

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

   if 1.05000000000000004 < f

   1. Initial program 11.0%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in f around inf 86.0%

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

      1. expm1-define 86.0%

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

      2. *-commutative 86.0%

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

      3. associate-*l* 86.0%

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

      4. expm1-define 86.0%

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

      5. associate-*r* 86.0%

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

      6. *-commutative 86.0%

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

      7. *-commutative 86.0%

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

   7. Simplified 86.0%

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

   8. Taylor expanded in f around 0 17.6%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\color{blue}{f \cdot \left(0.5 \cdot \pi + f \cdot \left(0.020833333333333332 \cdot \left(f \cdot {\pi}^{3}\right) + 0.125 \cdot {\pi}^{2}\right)\right)}} - \frac{1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi} \]

   9. Taylor expanded in f around inf 77.8%

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

       1. distribute-neg-frac 77.8%

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

       2. metadata-eval 77.8%

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

       3. expm1-define 77.8%

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

       4. *-commutative 77.8%

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

       5. *-commutative 77.8%

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

       6. associate-*r* 77.8%

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

   11. Simplified 77.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.05:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{f \cdot \pi}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.25)
		tmp = Float64(-4.0 * Float64(log(Float64(4.0 / Float64(f * pi))) / pi));
	else
		tmp = 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 / (f * pi))) / pi);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if f < 1.25

   1. Initial program 6.0%

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

   3. Simplified 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 inf 3.1%

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

      1. expm1-define 3.2%

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

      2. *-commutative 3.2%

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

      3. associate-*l* 3.2%

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

      4. expm1-define 99.5%

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

      5. associate-*r* 99.5%

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

      6. *-commutative 99.5%

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

      7. *-commutative 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in f around 0 98.6%

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

      1. *-commutative 98.6%

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

   10. Simplified 98.6%

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

   if 1.25 < f

   1. Initial program 11.0%

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

   3. Simplified 86.0%

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

   6. Applied egg-rr 1.3%

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

      1. +-inverses 1.3%

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

   8. Simplified 1.3%

      \[\leadsto \log \color{blue}{0} \cdot \frac{-4}{\pi} \]
   9. Step-by-step derivation

      1. +-inverses 1.3%

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

      2. sub-neg 1.3%

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

      3. add-sqr-sqrt 1.3%

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

      4. sqrt-unprod 1.3%

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

      5. *-commutative 1.3%

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

      6. *-commutative 1.3%

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

      7. frac-times 1.3%

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

      8. metadata-eval 1.3%

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

      9. metadata-eval 1.3%

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

      10. frac-times 1.3%

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

   10. Applied egg-rr 0.0%

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

       1. +-inverses 75.8%

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

   12. Simplified 75.8%

       \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]
   13. Step-by-step derivation

       1. mul0-lft 75.8%

          \[\leadsto \color{blue}{0} \]

   14. Applied egg-rr 75.8%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

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

Alternative 7: 7.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 230:\\ \;\;\;\;\frac{-16}{f \cdot {\pi}^{2}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 230.0) (/ -16.0 (* f (pow PI 2.0))) 0.0))

C

double code(double f) {
	double tmp;
	if (f <= 230.0) {
		tmp = -16.0 / (f * pow(((double) M_PI), 2.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 230.0) {
		tmp = -16.0 / (f * Math.pow(Math.PI, 2.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 230.0:
		tmp = -16.0 / (f * math.pow(math.pi, 2.0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 230.0)
		tmp = Float64(-16.0 / Float64(f * (pi ^ 2.0)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 230.0)
		tmp = -16.0 / (f * (pi ^ 2.0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 230.0], N[(-16.0 / N[(f * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 230

   1. Initial program 6.3%

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

   3. Simplified 98.9%

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

   6. Applied egg-rr 76.8%

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

      1. count-2 76.8%

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

      2. associate-*r/ 76.8%

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

      3. metadata-eval 76.8%

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

      4. *-commutative 76.8%

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

   8. Simplified 76.8%

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

   9. Taylor expanded in f around 0 76.8%

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

       1. associate-*r/ 76.8%

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

       2. metadata-eval 76.8%

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

   11. Simplified 76.8%

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

   12. Taylor expanded in f around inf 5.4%

       \[\leadsto \color{blue}{\frac{-16}{f \cdot {\pi}^{2}}} \]

   if 230 < 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 1.6%

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

      1. +-inverses 1.6%

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

   8. Simplified 1.6%

      \[\leadsto \log \color{blue}{0} \cdot \frac{-4}{\pi} \]
   9. Step-by-step derivation

      1. +-inverses 1.6%

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

      2. sub-neg 1.6%

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

      3. add-sqr-sqrt 1.6%

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

      4. sqrt-unprod 1.6%

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

      5. *-commutative 1.6%

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

      6. *-commutative 1.6%

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

      7. frac-times 1.6%

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

      8. metadata-eval 1.6%

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

      9. metadata-eval 1.6%

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

      10. frac-times 1.6%

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

   10. Applied egg-rr 0.0%

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

       1. +-inverses 100.0%

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

   12. Simplified 100.0%

       \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]
   13. Step-by-step derivation

       1. mul0-lft 100.0%

          \[\leadsto \color{blue}{0} \]

   14. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 4.9% accurate, 532.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (f) :precision binary64 0.0)

C

double code(double f) {
	return 0.0;
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = 0.0d0
end function

Java

public static double code(double f) {
	return 0.0;
}

Python

def code(f):
	return 0.0

Julia

function code(f)
	return 0.0
end

MATLAB

function tmp = code(f)
	tmp = 0.0;
end

Wolfram

code[f_] := 0.0

Derivation

1. Initial program 6.2%

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

3. Simplified 99.0%

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

6. Applied egg-rr 0.7%

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

   1. +-inverses 0.7%

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

8. Simplified 0.7%

   \[\leadsto \log \color{blue}{0} \cdot \frac{-4}{\pi} \]
9. Step-by-step derivation

   1. +-inverses 0.7%

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

   2. sub-neg 0.7%

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

   3. add-sqr-sqrt 4.9%

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

   4. sqrt-unprod 0.3%

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

   5. *-commutative 0.3%

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

   6. *-commutative 0.3%

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

   7. frac-times 1.4%

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

   8. metadata-eval 1.4%

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

   9. metadata-eval 1.4%

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

   10. frac-times 0.3%

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

10. Applied egg-rr 0.0%

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

    1. +-inverses 5.4%

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

12. Simplified 5.4%

    \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]
13. Step-by-step derivation

    1. mul0-lft 5.4%

       \[\leadsto \color{blue}{0} \]

14. Applied egg-rr 5.4%

    \[\leadsto \color{blue}{0} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024188 
(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)))))))))