VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.0%

Time: 31.8s

Alternatives: 9

Speedup: 4.9×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t\_0}\\ t_2 := e^{-t\_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t\_1 + t\_2}{t\_1 - t\_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Initial Program: 6.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t\_0}\\ t_2 := e^{-t\_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t\_1 + t\_2}{t\_1 - t\_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Alternative 1: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around inf 6.5%

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

7. Simplified 99.3%

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

8. Final simplification 99.3%

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

Alternative 2: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 225.0)
   (-
    (* -4.0 (/ (- (log (/ 4.0 PI)) (log f)) PI))
    (* (pow f 2.0) (* PI 0.08333333333333333)))
   (* -4.0 (/ (log (/ -1.0 (expm1 (* PI (* f -0.5))))) PI))))

C

double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = (-4.0 * ((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI))) - (pow(f, 2.0) * (((double) M_PI) * 0.08333333333333333));
	} else {
		tmp = -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 <= 225.0) {
		tmp = (-4.0 * ((Math.log((4.0 / Math.PI)) - Math.log(f)) / Math.PI)) - (Math.pow(f, 2.0) * (Math.PI * 0.08333333333333333));
	} else {
		tmp = -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 <= 225.0:
		tmp = (-4.0 * ((math.log((4.0 / math.pi)) - math.log(f)) / math.pi)) - (math.pow(f, 2.0) * (math.pi * 0.08333333333333333))
	else:
		tmp = -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 <= 225.0)
		tmp = Float64(Float64(-4.0 * Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi)) - Float64((f ^ 2.0) * Float64(pi * 0.08333333333333333)));
	else
		tmp = 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, 225.0], N[(N[(-4.0 * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] - N[(N[Power[f, 2.0], $MachinePrecision] * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 < 225

   1. Initial program 6.7%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in f around 0 98.9%

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

      1. mul-1-neg 98.9%

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

      2. unsub-neg 98.9%

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

      3. mul-1-neg 98.9%

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

      4. unsub-neg 98.9%

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

      5. distribute-rgt-out 98.9%

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

      6. distribute-rgt-out 98.9%

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

   7. Simplified 98.9%

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

   if 225 < f

   1. Initial program 5.6%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in f around inf 93.1%

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

   7. Simplified 93.1%

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

   8. Taylor expanded in f around 0 3.2%

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

   9. Taylor expanded in f around inf 93.1%

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

       1. distribute-neg-frac 93.1%

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

       2. metadata-eval 93.1%

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

       3. expm1-define 93.1%

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

       4. *-commutative 93.1%

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

       5. *-commutative 93.1%

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

       6. associate-*l* 93.1%

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

   11. Simplified 93.1%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\\ \mathbf{if}\;f \leq 225:\\ \;\;\;\;-4 \cdot \frac{\log \left(t\_0 + \frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + f \cdot \left(\pi \cdot 0.08333333333333333 + \pi \cdot -0.125\right)\right)}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\log t\_0}{\pi}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ -1.0 (expm1 (* PI (* f -0.5))))))
   (if (<= f 225.0)
     (*
      -4.0
      (/
       (log
        (+
         t_0
         (/
          (-
           (* 2.0 (/ 1.0 PI))
           (* f (+ 0.5 (* f (+ (* PI 0.08333333333333333) (* PI -0.125))))))
          f)))
       PI))
     (* -4.0 (/ (log t_0) PI)))))

C

double code(double f) {
	double t_0 = -1.0 / expm1((((double) M_PI) * (f * -0.5)));
	double tmp;
	if (f <= 225.0) {
		tmp = -4.0 * (log((t_0 + (((2.0 * (1.0 / ((double) M_PI))) - (f * (0.5 + (f * ((((double) M_PI) * 0.08333333333333333) + (((double) M_PI) * -0.125)))))) / f))) / ((double) M_PI));
	} else {
		tmp = -4.0 * (log(t_0) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = -1.0 / Math.expm1((Math.PI * (f * -0.5)));
	double tmp;
	if (f <= 225.0) {
		tmp = -4.0 * (Math.log((t_0 + (((2.0 * (1.0 / Math.PI)) - (f * (0.5 + (f * ((Math.PI * 0.08333333333333333) + (Math.PI * -0.125)))))) / f))) / Math.PI);
	} else {
		tmp = -4.0 * (Math.log(t_0) / Math.PI);
	}
	return tmp;
}

Python

def code(f):
	t_0 = -1.0 / math.expm1((math.pi * (f * -0.5)))
	tmp = 0
	if f <= 225.0:
		tmp = -4.0 * (math.log((t_0 + (((2.0 * (1.0 / math.pi)) - (f * (0.5 + (f * ((math.pi * 0.08333333333333333) + (math.pi * -0.125)))))) / f))) / math.pi)
	else:
		tmp = -4.0 * (math.log(t_0) / math.pi)
	return tmp

Julia

function code(f)
	t_0 = Float64(-1.0 / expm1(Float64(pi * Float64(f * -0.5))))
	tmp = 0.0
	if (f <= 225.0)
		tmp = Float64(-4.0 * Float64(log(Float64(t_0 + Float64(Float64(Float64(2.0 * Float64(1.0 / pi)) - Float64(f * Float64(0.5 + Float64(f * Float64(Float64(pi * 0.08333333333333333) + Float64(pi * -0.125)))))) / f))) / pi));
	else
		tmp = Float64(-4.0 * Float64(log(t_0) / pi));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[(-1.0 / N[(Exp[N[(Pi * N[(f * -0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[f, 225.0], N[(-4.0 * N[(N[Log[N[(t$95$0 + N[(N[(N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] - N[(f * N[(0.5 + N[(f * N[(N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(Pi * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[Log[t$95$0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if f < 225

   1. Initial program 6.7%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in f around inf 3.7%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in f around 0 98.9%

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

   if 225 < f

   1. Initial program 5.6%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in f around inf 93.1%

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

   7. Simplified 93.1%

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

   8. Taylor expanded in f around 0 3.2%

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

   9. Taylor expanded in f around inf 93.1%

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

       1. distribute-neg-frac 93.1%

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

       2. metadata-eval 93.1%

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

       3. expm1-define 93.1%

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

       4. *-commutative 93.1%

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

       5. *-commutative 93.1%

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

       6. associate-*l* 93.1%

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

   11. Simplified 93.1%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} + \frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + f \cdot \left(\pi \cdot 0.08333333333333333 + \pi \cdot -0.125\right)\right)}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 225.0)
   (-
    (* -4.0 (/ (log (/ (/ 4.0 PI) f)) PI))
    (* (pow f 2.0) (* PI 0.08333333333333333)))
   (* -4.0 (/ (log (/ -1.0 (expm1 (* PI (* f -0.5))))) PI))))

C

double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = (-4.0 * (log(((4.0 / ((double) M_PI)) / f)) / ((double) M_PI))) - (pow(f, 2.0) * (((double) M_PI) * 0.08333333333333333));
	} 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 <= 225.0) {
		tmp = (-4.0 * (Math.log(((4.0 / Math.PI) / f)) / Math.PI)) - (Math.pow(f, 2.0) * (Math.PI * 0.08333333333333333));
	} 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 <= 225.0:
		tmp = (-4.0 * (math.log(((4.0 / math.pi) / f)) / math.pi)) - (math.pow(f, 2.0) * (math.pi * 0.08333333333333333))
	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 <= 225.0)
		tmp = Float64(Float64(-4.0 * Float64(log(Float64(Float64(4.0 / pi) / f)) / pi)) - Float64((f ^ 2.0) * Float64(pi * 0.08333333333333333)));
	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, 225.0], N[(N[(-4.0 * N[(N[Log[N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] - N[(N[Power[f, 2.0], $MachinePrecision] * N[(Pi * 0.08333333333333333), $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 < 225

   1. Initial program 6.7%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in f around 0 98.9%

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

      1. mul-1-neg 98.9%

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

      2. unsub-neg 98.9%

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

      3. mul-1-neg 98.9%

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

      4. unsub-neg 98.9%

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

      5. distribute-rgt-out 98.9%

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

      6. distribute-rgt-out 98.9%

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

   7. Simplified 98.9%

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

      1. *-un-lft-identity 98.1%

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

      2. diff-log 98.0%

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

   9. Applied egg-rr 98.9%

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

       1. *-lft-identity 98.0%

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

   11. Simplified 98.9%

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

   if 225 < f

   1. Initial program 5.6%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in f around inf 93.1%

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

   7. Simplified 93.1%

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

   8. Taylor expanded in f around 0 3.2%

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

   9. Taylor expanded in f around inf 93.1%

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

       1. distribute-neg-frac 93.1%

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

       2. metadata-eval 93.1%

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

       3. expm1-define 93.1%

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

       4. *-commutative 93.1%

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

       5. *-commutative 93.1%

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

       6. associate-*l* 93.1%

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

   11. Simplified 93.1%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

double code(double f) {
	double tmp;
	if (f <= 1.0) {
		tmp = -4.0 * ((log((4.0 / ((double) M_PI))) - 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 <= 1.0) {
		tmp = -4.0 * ((Math.log((4.0 / Math.PI)) - 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 <= 1.0:
		tmp = -4.0 * ((math.log((4.0 / math.pi)) - 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 <= 1.0)
		tmp = Float64(-4.0 * Float64(Float64(log(Float64(4.0 / pi)) - 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, 1.0], N[(-4.0 * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $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

   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%

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

   5. Taylor expanded in f around 0 98.5%

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

      1. mul-1-neg 98.5%

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

      2. unsub-neg 98.5%

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

   7. Simplified 98.5%

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

   if 1 < f

   1. Initial program 16.1%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in f around inf 93.9%

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

   7. Simplified 93.9%

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

   8. Taylor expanded in f around 0 4.9%

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

   9. Taylor expanded in f around inf 84.9%

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

       1. distribute-neg-frac 84.9%

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

       2. metadata-eval 84.9%

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

       3. expm1-define 84.9%

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

       4. *-commutative 84.9%

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

       5. *-commutative 84.9%

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

       6. associate-*l* 84.9%

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

   11. Simplified 84.9%

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

4. Final simplification 98.0%

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

Alternative 6: 96.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around 0 95.1%

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

   1. mul-1-neg 95.1%

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

   2. unsub-neg 95.1%

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

7. Simplified 95.1%

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

8. Final simplification 95.1%

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

Alternative 7: 1.6% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around 0 95.1%

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

   1. mul-1-neg 95.1%

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

   2. unsub-neg 95.1%

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

7. Simplified 95.1%

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

   1. expm1-log1p-u 93.7%

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

   2. expm1-undefine 93.7%

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

   3. diff-log 93.7%

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

9. Applied egg-rr 93.7%

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

    1. expm1-define 93.7%

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

11. Simplified 93.7%

    \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}\right)\right)} \]
12. Step-by-step derivation

    1. expm1-log1p-u 95.1%

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

    2. add-sqr-sqrt 94.4%

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

    3. sqrt-unprod 95.1%

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

    4. clear-num 95.1%

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

    5. neg-log 95.1%

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

    6. clear-num 95.1%

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

    7. neg-log 95.1%

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

    8. sqr-neg 95.1%

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

    9. sqrt-unprod 0.2%

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

    10. add-sqr-sqrt 1.7%

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

    11. associate-/r/ 1.7%

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

13. Applied egg-rr 1.7%

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

14. Final simplification 1.7%

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

Alternative 8: 96.1% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around 0 95.1%

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

   1. mul-1-neg 95.1%

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

   2. unsub-neg 95.1%

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

7. Simplified 95.1%

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

   1. *-un-lft-identity 95.1%

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

   2. diff-log 95.1%

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

9. Applied egg-rr 95.1%

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

    1. *-lft-identity 95.1%

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

11. Simplified 95.1%

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

12. Final simplification 95.1%

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

Alternative 9: 1.6% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log 0.5}{\pi} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(f)
	tmp = -4.0 * (log(0.5) / pi);
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[0.5], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around inf 6.5%

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

7. Simplified 99.3%

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

8. Taylor expanded in f around 0 94.5%

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

9. Taylor expanded in f around 0 94.5%

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

10. Taylor expanded in f around inf 1.6%

    \[\leadsto -4 \cdot \color{blue}{\frac{\log 0.5}{\pi}} \]

11. Final simplification 1.6%

    \[\leadsto -4 \cdot \frac{\log 0.5}{\pi} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024058 
(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)))))))))