VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 98.8%

Time: 16.8s

Alternatives: 11

Speedup: 33.2×

• could not determine a ground truth

  1. f = 1349851138.8584216

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: 98.8% 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 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 99.3%

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

5. Final simplification 99.3%

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

Alternative 2: 98.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 220:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 220.0)
   (-
    (* -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 <= 220.0) {
		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 <= 220.0) {
		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 <= 220.0:
		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 <= 220.0)
		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 <= 220.0)
		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, 220.0], 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 < 220

   1. Initial program 6.4%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in f around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. unsub-neg 98.7%

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

      3. mul-1-neg 98.7%

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

      4. unsub-neg 98.7%

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

      5. distribute-rgt-out 98.7%

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

      6. metadata-eval 98.7%

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

   7. Simplified 98.7%

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

   if 220 < 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%

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

      1. flip-+ 0.0%

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

      2. log-div 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

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

      2. div0 0.0%

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

      3. +-inverses 100.0%

         \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]
   11. Step-by-step derivation

       1. add-log-exp 100.0%

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

       2. metadata-eval 100.0%

          \[\leadsto \log \left(e^{\frac{1}{\frac{\pi}{\color{blue}{0}}}}\right) \]

       3. associate-/r/ 100.0%

          \[\leadsto \log \left(e^{\color{blue}{\frac{1}{\pi} \cdot 0}}\right) \]

       4. exp-prod 100.0%

          \[\leadsto \log \color{blue}{\left({\left(e^{\frac{1}{\pi}}\right)}^{0}\right)} \]

       5. metadata-eval 100.0%

          \[\leadsto \log \color{blue}{1} \]

       6. metadata-eval 100.0%

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

   12. Applied egg-rr 100.0%

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

Alternative 3: 98.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 230:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{f \cdot \left(0.5 + f \cdot \left(\pi \cdot 0.041666666666666664\right)\right) + \frac{2}{\pi}}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 230.0)
   (*
    (/ -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 <= 230.0) {
		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 <= 230.0) {
		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 <= 230.0:
		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 <= 230.0)
		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, 230.0], 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 < 230

   1. Initial program 6.4%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in f around 0 98.6%

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

      1. pow1 98.6%

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

      2. mul-1-neg 98.6%

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

      3. distribute-rgt-out 98.6%

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

      4. metadata-eval 98.6%

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

   7. Applied egg-rr 98.6%

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

      1. unpow1 98.6%

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

      2. distribute-rgt-neg-in 98.6%

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

      3. distribute-rgt-neg-in 98.6%

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

      4. metadata-eval 98.6%

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

   9. Simplified 98.6%

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

       1. un-div-inv 98.6%

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

   11. Applied egg-rr 98.6%

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

   if 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 3.1%

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

      1. flip-+ 0.0%

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

      2. log-div 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

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

      2. div0 0.0%

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

      3. +-inverses 100.0%

         \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]
   11. Step-by-step derivation

       1. add-log-exp 100.0%

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

       2. metadata-eval 100.0%

          \[\leadsto \log \left(e^{\frac{1}{\frac{\pi}{\color{blue}{0}}}}\right) \]

       3. associate-/r/ 100.0%

          \[\leadsto \log \left(e^{\color{blue}{\frac{1}{\pi} \cdot 0}}\right) \]

       4. exp-prod 100.0%

          \[\leadsto \log \color{blue}{\left({\left(e^{\frac{1}{\pi}}\right)}^{0}\right)} \]

       5. metadata-eval 100.0%

          \[\leadsto \log \color{blue}{1} \]

       6. metadata-eval 100.0%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 230:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{f \cdot \left(0.5 + f \cdot \left(\pi \cdot 0.041666666666666664\right)\right) + \frac{2}{\pi}}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.26) (* -4.0 (/ (- (log (/ 4.0 PI)) (log f)) PI)) 0.0))

C

double code(double f) {
	double tmp;
	if (f <= 1.26) {
		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 <= 1.26) {
		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 <= 1.26:
		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 <= 1.26)
		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 <= 1.26)
		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, 1.26], 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 < 1.26000000000000001

   1. Initial program 6.1%

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

   3. Simplified 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 98.7%

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

      1. mul-1-neg 98.7%

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

      2. unsub-neg 98.7%

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

   7. Simplified 98.7%

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

   if 1.26000000000000001 < f

   1. Initial program 12.3%

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

   3. Simplified 95.7%

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

   6. Applied egg-rr 4.7%

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

      1. flip-+ 0.0%

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

      2. log-div 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

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

      2. div0 0.0%

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

      3. +-inverses 83.9%

         \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]

   10. Simplified 83.9%

       \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]
   11. Step-by-step derivation

       1. add-log-exp 83.9%

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

       2. metadata-eval 83.9%

          \[\leadsto \log \left(e^{\frac{1}{\frac{\pi}{\color{blue}{0}}}}\right) \]

       3. associate-/r/ 83.9%

          \[\leadsto \log \left(e^{\color{blue}{\frac{1}{\pi} \cdot 0}}\right) \]

       4. exp-prod 83.9%

          \[\leadsto \log \color{blue}{\left({\left(e^{\frac{1}{\pi}}\right)}^{0}\right)} \]

       5. metadata-eval 83.9%

          \[\leadsto \log \color{blue}{1} \]

       6. metadata-eval 83.9%

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

   12. Applied egg-rr 83.9%

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

Alternative 5: 97.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \frac{-1}{\pi}\\ \mathbf{if}\;f \leq 230:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{-0.5 \cdot f - t\_0}{f} - \frac{f \cdot \left(f \cdot \left(\pi \cdot 0.08333333333333333 + \pi \cdot -0.125\right) - 0.5\right) + t\_0}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* 2.0 (/ -1.0 PI))))
   (if (<= f 230.0)
     (*
      (/ -4.0 PI)
      (log
       (-
        (/ (- (* -0.5 f) t_0) f)
        (/
         (+
          (* f (- (* f (+ (* PI 0.08333333333333333) (* PI -0.125))) 0.5))
          t_0)
         f))))
     0.0)))

C

double code(double f) {
	double t_0 = 2.0 * (-1.0 / ((double) M_PI));
	double tmp;
	if (f <= 230.0) {
		tmp = (-4.0 / ((double) M_PI)) * log(((((-0.5 * f) - t_0) / f) - (((f * ((f * ((((double) M_PI) * 0.08333333333333333) + (((double) M_PI) * -0.125))) - 0.5)) + t_0) / f)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = 2.0 * (-1.0 / Math.PI);
	double tmp;
	if (f <= 230.0) {
		tmp = (-4.0 / Math.PI) * Math.log(((((-0.5 * f) - t_0) / f) - (((f * ((f * ((Math.PI * 0.08333333333333333) + (Math.PI * -0.125))) - 0.5)) + t_0) / f)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	t_0 = 2.0 * (-1.0 / math.pi)
	tmp = 0
	if f <= 230.0:
		tmp = (-4.0 / math.pi) * math.log(((((-0.5 * f) - t_0) / f) - (((f * ((f * ((math.pi * 0.08333333333333333) + (math.pi * -0.125))) - 0.5)) + t_0) / f)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	t_0 = Float64(2.0 * Float64(-1.0 / pi))
	tmp = 0.0
	if (f <= 230.0)
		tmp = Float64(Float64(-4.0 / pi) * log(Float64(Float64(Float64(Float64(-0.5 * f) - t_0) / f) - Float64(Float64(Float64(f * Float64(Float64(f * Float64(Float64(pi * 0.08333333333333333) + Float64(pi * -0.125))) - 0.5)) + t_0) / f))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	t_0 = 2.0 * (-1.0 / pi);
	tmp = 0.0;
	if (f <= 230.0)
		tmp = (-4.0 / pi) * log(((((-0.5 * f) - t_0) / f) - (((f * ((f * ((pi * 0.08333333333333333) + (pi * -0.125))) - 0.5)) + t_0) / f)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := Block[{t$95$0 = N[(2.0 * N[(-1.0 / Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[f, 230.0], N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(N[(N[(N[(-0.5 * f), $MachinePrecision] - t$95$0), $MachinePrecision] / f), $MachinePrecision] - N[(N[(N[(f * N[(N[(f * N[(N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(Pi * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if f < 230

   1. Initial program 6.4%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in f around 0 98.2%

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

   6. Taylor expanded in f around 0 98.3%

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

   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 3.1%

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

      1. flip-+ 0.0%

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

      2. log-div 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

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

      2. div0 0.0%

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

      3. +-inverses 100.0%

         \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]
   11. Step-by-step derivation

       1. add-log-exp 100.0%

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

       2. metadata-eval 100.0%

          \[\leadsto \log \left(e^{\frac{1}{\frac{\pi}{\color{blue}{0}}}}\right) \]

       3. associate-/r/ 100.0%

          \[\leadsto \log \left(e^{\color{blue}{\frac{1}{\pi} \cdot 0}}\right) \]

       4. exp-prod 100.0%

          \[\leadsto \log \color{blue}{\left({\left(e^{\frac{1}{\pi}}\right)}^{0}\right)} \]

       5. metadata-eval 100.0%

          \[\leadsto \log \color{blue}{1} \]

       6. metadata-eval 100.0%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 98.3%

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

Alternative 6: 97.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;\frac{-1}{\frac{-1}{\frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi \cdot -0.25}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.26)
   (/ -1.0 (/ -1.0 (/ (log (/ 4.0 (* PI f))) (* PI -0.25))))
   0.0))

C

double code(double f) {
	double tmp;
	if (f <= 1.26) {
		tmp = -1.0 / (-1.0 / (log((4.0 / (((double) M_PI) * f))) / (((double) M_PI) * -0.25)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.26) {
		tmp = -1.0 / (-1.0 / (Math.log((4.0 / (Math.PI * f))) / (Math.PI * -0.25)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.26:
		tmp = -1.0 / (-1.0 / (math.log((4.0 / (math.pi * f))) / (math.pi * -0.25)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.26)
		tmp = Float64(-1.0 / Float64(-1.0 / Float64(log(Float64(4.0 / Float64(pi * f))) / Float64(pi * -0.25))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.26)
		tmp = -1.0 / (-1.0 / (log((4.0 / (pi * f))) / (pi * -0.25)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.26], N[(-1.0 / N[(-1.0 / N[(N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 1.26000000000000001

   1. Initial program 6.1%

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

   3. Simplified 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 97.8%

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

   7. Taylor expanded in f around 0 98.6%

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

      1. associate-*r/ 98.6%

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

      2. neg-mul-1 98.6%

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

      3. unsub-neg 98.6%

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

   9. Simplified 98.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{-0.25 \cdot \pi}{\log \left(\frac{4}{\pi}\right) - \log f}}} \]
   10. Step-by-step derivation

       1. clear-num 98.6%

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

       2. inv-pow 98.6%

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

       3. diff-log 98.7%

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

       4. *-commutative 98.7%

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

   11. Applied egg-rr 98.7%

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

       1. unpow-1 98.7%

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

       2. associate-/l/ 98.7%

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

   13. Simplified 98.7%

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

   if 1.26000000000000001 < f

   1. Initial program 12.3%

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

   3. Simplified 95.7%

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

   6. Applied egg-rr 4.7%

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

      1. flip-+ 0.0%

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

      2. log-div 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

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

      2. div0 0.0%

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

      3. +-inverses 83.9%

         \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]

   10. Simplified 83.9%

       \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]
   11. Step-by-step derivation

       1. add-log-exp 83.9%

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

       2. metadata-eval 83.9%

          \[\leadsto \log \left(e^{\frac{1}{\frac{\pi}{\color{blue}{0}}}}\right) \]

       3. associate-/r/ 83.9%

          \[\leadsto \log \left(e^{\color{blue}{\frac{1}{\pi} \cdot 0}}\right) \]

       4. exp-prod 83.9%

          \[\leadsto \log \color{blue}{\left({\left(e^{\frac{1}{\pi}}\right)}^{0}\right)} \]

       5. metadata-eval 83.9%

          \[\leadsto \log \color{blue}{1} \]

       6. metadata-eval 83.9%

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

   12. Applied egg-rr 83.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;\frac{-1}{\frac{-1}{\frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi \cdot -0.25}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;\frac{1}{\frac{\pi \cdot -0.25}{\log \left(\frac{4}{\pi \cdot f}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.26) (/ 1.0 (/ (* PI -0.25) (log (/ 4.0 (* PI f))))) 0.0))

C

double code(double f) {
	double tmp;
	if (f <= 1.26) {
		tmp = 1.0 / ((((double) M_PI) * -0.25) / log((4.0 / (((double) M_PI) * f))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.26) {
		tmp = 1.0 / ((Math.PI * -0.25) / Math.log((4.0 / (Math.PI * f))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.26:
		tmp = 1.0 / ((math.pi * -0.25) / math.log((4.0 / (math.pi * f))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.26)
		tmp = Float64(1.0 / Float64(Float64(pi * -0.25) / log(Float64(4.0 / Float64(pi * f)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.26)
		tmp = 1.0 / ((pi * -0.25) / log((4.0 / (pi * f))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.26], N[(1.0 / N[(N[(Pi * -0.25), $MachinePrecision] / N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 1.26000000000000001

   1. Initial program 6.1%

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

   3. Simplified 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 97.8%

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

   7. Taylor expanded in f around 0 98.6%

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

      1. associate-*r/ 98.6%

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

      2. neg-mul-1 98.6%

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

      3. unsub-neg 98.6%

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

   9. Simplified 98.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{-0.25 \cdot \pi}{\log \left(\frac{4}{\pi}\right) - \log f}}} \]
   10. Step-by-step derivation

       1. inv-pow 98.6%

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

       2. *-commutative 98.6%

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

       3. diff-log 98.6%

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

   11. Applied egg-rr 98.6%

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

       1. unpow-1 98.6%

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

       2. associate-/l/ 98.6%

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

   13. Simplified 98.6%

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

   if 1.26000000000000001 < f

   1. Initial program 12.3%

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

   3. Simplified 95.7%

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

   6. Applied egg-rr 4.7%

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

      1. flip-+ 0.0%

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

      2. log-div 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

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

      2. div0 0.0%

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

      3. +-inverses 83.9%

         \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]

   10. Simplified 83.9%

       \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]
   11. Step-by-step derivation

       1. add-log-exp 83.9%

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

       2. metadata-eval 83.9%

          \[\leadsto \log \left(e^{\frac{1}{\frac{\pi}{\color{blue}{0}}}}\right) \]

       3. associate-/r/ 83.9%

          \[\leadsto \log \left(e^{\color{blue}{\frac{1}{\pi} \cdot 0}}\right) \]

       4. exp-prod 83.9%

          \[\leadsto \log \color{blue}{\left({\left(e^{\frac{1}{\pi}}\right)}^{0}\right)} \]

       5. metadata-eval 83.9%

          \[\leadsto \log \color{blue}{1} \]

       6. metadata-eval 83.9%

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

   12. Applied egg-rr 83.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;\frac{1}{\frac{\pi \cdot -0.25}{\log \left(\frac{4}{\pi \cdot f}\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{4}{\pi \cdot f}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.26) (* (/ -4.0 PI) (log (/ 4.0 (* PI f)))) 0.0))

C

double code(double f) {
	double tmp;
	if (f <= 1.26) {
		tmp = (-4.0 / ((double) M_PI)) * log((4.0 / (((double) M_PI) * f)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.26) {
		tmp = (-4.0 / Math.PI) * Math.log((4.0 / (Math.PI * f)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.26:
		tmp = (-4.0 / math.pi) * math.log((4.0 / (math.pi * f)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.26)
		tmp = Float64(Float64(-4.0 / pi) * log(Float64(4.0 / Float64(pi * f))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.26)
		tmp = (-4.0 / pi) * log((4.0 / (pi * f)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.26], N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 1.26000000000000001

   1. Initial program 6.1%

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

   3. Simplified 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 98.6%

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

      1. *-commutative 98.6%

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

   7. Simplified 98.6%

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

   if 1.26000000000000001 < f

   1. Initial program 12.3%

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

   3. Simplified 95.7%

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

   6. Applied egg-rr 4.7%

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

      1. flip-+ 0.0%

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

      2. log-div 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

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

      2. div0 0.0%

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

      3. +-inverses 83.9%

         \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]

   10. Simplified 83.9%

       \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]
   11. Step-by-step derivation

       1. add-log-exp 83.9%

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

       2. metadata-eval 83.9%

          \[\leadsto \log \left(e^{\frac{1}{\frac{\pi}{\color{blue}{0}}}}\right) \]

       3. associate-/r/ 83.9%

          \[\leadsto \log \left(e^{\color{blue}{\frac{1}{\pi} \cdot 0}}\right) \]

       4. exp-prod 83.9%

          \[\leadsto \log \color{blue}{\left({\left(e^{\frac{1}{\pi}}\right)}^{0}\right)} \]

       5. metadata-eval 83.9%

          \[\leadsto \log \color{blue}{1} \]

       6. metadata-eval 83.9%

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

   12. Applied egg-rr 83.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{4}{\pi \cdot f}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 16.0% accurate, 26.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 230:\\ \;\;\;\;\frac{-1}{\frac{\pi}{-4} \cdot \left(f \cdot \left(\pi \cdot -0.25 - \frac{0.5}{f}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 230.0)
   (/ -1.0 (* (/ PI -4.0) (* f (- (* PI -0.25) (/ 0.5 f)))))
   0.0))

C

double code(double f) {
	double tmp;
	if (f <= 230.0) {
		tmp = -1.0 / ((((double) M_PI) / -4.0) * (f * ((((double) M_PI) * -0.25) - (0.5 / f))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 230.0) {
		tmp = -1.0 / ((Math.PI / -4.0) * (f * ((Math.PI * -0.25) - (0.5 / f))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 230.0:
		tmp = -1.0 / ((math.pi / -4.0) * (f * ((math.pi * -0.25) - (0.5 / f))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 230.0)
		tmp = Float64(-1.0 / Float64(Float64(pi / -4.0) * Float64(f * Float64(Float64(pi * -0.25) - Float64(0.5 / f)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 230.0)
		tmp = -1.0 / ((pi / -4.0) * (f * ((pi * -0.25) - (0.5 / f))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 230.0], N[(-1.0 / N[(N[(Pi / -4.0), $MachinePrecision] * N[(f * N[(N[(Pi * -0.25), $MachinePrecision] - N[(0.5 / f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 230

   1. Initial program 6.4%

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

   3. Simplified 99.3%

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

   6. Applied egg-rr 97.5%

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

      1. *-un-lft-identity 97.5%

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

      2. times-frac 97.4%

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

      3. count-2 97.4%

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

      4. *-commutative 97.4%

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

   8. Applied egg-rr 97.4%

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

   9. Taylor expanded in f around 0 97.4%

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

       1. associate-*r/ 97.4%

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

       2. metadata-eval 97.4%

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

   11. Simplified 97.4%

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

   12. Taylor expanded in f around -inf 14.4%

       \[\leadsto \frac{1}{\color{blue}{\left(-1 \cdot \left(f \cdot \left(-0.25 \cdot \pi - 0.5 \cdot \frac{1}{f}\right)\right)\right)} \cdot \frac{\pi}{-4}} \]
   13. Step-by-step derivation

       1. associate-*r* 14.4%

          \[\leadsto \frac{1}{\color{blue}{\left(\left(-1 \cdot f\right) \cdot \left(-0.25 \cdot \pi - 0.5 \cdot \frac{1}{f}\right)\right)} \cdot \frac{\pi}{-4}} \]

       2. neg-mul-1 14.4%

          \[\leadsto \frac{1}{\left(\color{blue}{\left(-f\right)} \cdot \left(-0.25 \cdot \pi - 0.5 \cdot \frac{1}{f}\right)\right) \cdot \frac{\pi}{-4}} \]

       3. *-commutative 14.4%

          \[\leadsto \frac{1}{\left(\left(-f\right) \cdot \left(\color{blue}{\pi \cdot -0.25} - 0.5 \cdot \frac{1}{f}\right)\right) \cdot \frac{\pi}{-4}} \]

       4. associate-*r/ 14.4%

          \[\leadsto \frac{1}{\left(\left(-f\right) \cdot \left(\pi \cdot -0.25 - \color{blue}{\frac{0.5 \cdot 1}{f}}\right)\right) \cdot \frac{\pi}{-4}} \]

       5. metadata-eval 14.4%

          \[\leadsto \frac{1}{\left(\left(-f\right) \cdot \left(\pi \cdot -0.25 - \frac{\color{blue}{0.5}}{f}\right)\right) \cdot \frac{\pi}{-4}} \]

   14. Simplified 14.4%

       \[\leadsto \frac{1}{\color{blue}{\left(\left(-f\right) \cdot \left(\pi \cdot -0.25 - \frac{0.5}{f}\right)\right)} \cdot \frac{\pi}{-4}} \]

   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 3.1%

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

      1. flip-+ 0.0%

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

      2. log-div 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

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

      2. div0 0.0%

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

      3. +-inverses 100.0%

         \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]
   11. Step-by-step derivation

       1. add-log-exp 100.0%

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

       2. metadata-eval 100.0%

          \[\leadsto \log \left(e^{\frac{1}{\frac{\pi}{\color{blue}{0}}}}\right) \]

       3. associate-/r/ 100.0%

          \[\leadsto \log \left(e^{\color{blue}{\frac{1}{\pi} \cdot 0}}\right) \]

       4. exp-prod 100.0%

          \[\leadsto \log \color{blue}{\left({\left(e^{\frac{1}{\pi}}\right)}^{0}\right)} \]

       5. metadata-eval 100.0%

          \[\leadsto \log \color{blue}{1} \]

       6. metadata-eval 100.0%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 16.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 230:\\ \;\;\;\;\frac{-1}{\frac{\pi}{-4} \cdot \left(f \cdot \left(\pi \cdot -0.25 - \frac{0.5}{f}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 7.3% accurate, 33.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 230:\\ \;\;\;\;\frac{1}{\frac{\pi}{-4} \cdot \left(\left(\pi \cdot f\right) \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 230.0) (/ 1.0 (* (/ PI -4.0) (* (* PI f) 0.25))) 0.0))

C

double code(double f) {
	double tmp;
	if (f <= 230.0) {
		tmp = 1.0 / ((((double) M_PI) / -4.0) * ((((double) M_PI) * f) * 0.25));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 230.0) {
		tmp = 1.0 / ((Math.PI / -4.0) * ((Math.PI * f) * 0.25));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 230.0:
		tmp = 1.0 / ((math.pi / -4.0) * ((math.pi * f) * 0.25))
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 230.0)
		tmp = Float64(1.0 / Float64(Float64(pi / -4.0) * Float64(Float64(pi * f) * 0.25)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 230.0)
		tmp = 1.0 / ((pi / -4.0) * ((pi * f) * 0.25));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 230.0], N[(1.0 / N[(N[(Pi / -4.0), $MachinePrecision] * N[(N[(Pi * f), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 230

   1. Initial program 6.4%

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

   3. Simplified 99.3%

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

   6. Applied egg-rr 97.5%

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

      1. *-un-lft-identity 97.5%

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

      2. times-frac 97.4%

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

      3. count-2 97.4%

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

      4. *-commutative 97.4%

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

   8. Applied egg-rr 97.4%

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

   9. Taylor expanded in f around 0 97.4%

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

       1. associate-*r/ 97.4%

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

       2. metadata-eval 97.4%

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

   11. Simplified 97.4%

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

   12. Taylor expanded in f around inf 5.3%

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

       1. *-commutative 5.3%

          \[\leadsto \frac{1}{\color{blue}{\left(\left(f \cdot \pi\right) \cdot 0.25\right)} \cdot \frac{\pi}{-4}} \]

   14. Simplified 5.3%

       \[\leadsto \frac{1}{\color{blue}{\left(\left(f \cdot \pi\right) \cdot 0.25\right)} \cdot \frac{\pi}{-4}} \]

   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 3.1%

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

      1. flip-+ 0.0%

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

      2. log-div 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

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

      2. div0 0.0%

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

      3. +-inverses 100.0%

         \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]
   11. Step-by-step derivation

       1. add-log-exp 100.0%

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

       2. metadata-eval 100.0%

          \[\leadsto \log \left(e^{\frac{1}{\frac{\pi}{\color{blue}{0}}}}\right) \]

       3. associate-/r/ 100.0%

          \[\leadsto \log \left(e^{\color{blue}{\frac{1}{\pi} \cdot 0}}\right) \]

       4. exp-prod 100.0%

          \[\leadsto \log \color{blue}{\left({\left(e^{\frac{1}{\pi}}\right)}^{0}\right)} \]

       5. metadata-eval 100.0%

          \[\leadsto \log \color{blue}{1} \]

       6. metadata-eval 100.0%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 7.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 230:\\ \;\;\;\;\frac{1}{\frac{\pi}{-4} \cdot \left(\left(\pi \cdot f\right) \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 6.3%

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

3. Simplified 99.3%

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

6. Applied egg-rr 95.6%

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

   1. flip-+ 0.0%

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

   2. log-div 0.0%

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

8. Applied egg-rr 0.0%

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

   1. +-inverses 0.0%

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

   2. div0 0.0%

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

   3. +-inverses 5.0%

      \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]

10. Simplified 5.0%

    \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{0} \cdot -4}} \]
11. Step-by-step derivation

    1. add-log-exp 5.0%

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

    2. metadata-eval 5.0%

       \[\leadsto \log \left(e^{\frac{1}{\frac{\pi}{\color{blue}{0}}}}\right) \]

    3. associate-/r/ 5.0%

       \[\leadsto \log \left(e^{\color{blue}{\frac{1}{\pi} \cdot 0}}\right) \]

    4. exp-prod 5.0%

       \[\leadsto \log \color{blue}{\left({\left(e^{\frac{1}{\pi}}\right)}^{0}\right)} \]

    5. metadata-eval 5.0%

       \[\leadsto \log \color{blue}{1} \]

    6. metadata-eval 5.0%

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

12. Applied egg-rr 5.0%

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

Reproduce

herbie shell --seed 2024181 
(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)))))))))