VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 99.0%

Time: 33.2s

Alternatives: 5

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: 6.6% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 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 98.9%

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

5. Taylor expanded in f around inf 5.5%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

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

Alternative 2: 98.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)\\ \mathbf{if}\;f \leq 225:\\ \;\;\;\;\log \left(\frac{-1}{t\_0} + \frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + \pi \cdot \left(f \cdot -0.041666666666666664\right)\right)}{f}\right) \cdot \frac{-4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\log \left(-t\_0\right) \cdot \frac{4}{\pi}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (expm1 (* PI (* f -0.5)))))
   (if (<= f 225.0)
     (*
      (log
       (+
        (/ -1.0 t_0)
        (/
         (-
          (* 2.0 (/ 1.0 PI))
          (* f (+ 0.5 (* PI (* f -0.041666666666666664)))))
         f)))
      (/ -4.0 PI))
     (* (log (- t_0)) (/ 4.0 PI)))))

C

double code(double f) {
	double t_0 = expm1((((double) M_PI) * (f * -0.5)));
	double tmp;
	if (f <= 225.0) {
		tmp = log(((-1.0 / t_0) + (((2.0 * (1.0 / ((double) M_PI))) - (f * (0.5 + (((double) M_PI) * (f * -0.041666666666666664))))) / f))) * (-4.0 / ((double) M_PI));
	} else {
		tmp = log(-t_0) * (4.0 / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = Math.expm1((Math.PI * (f * -0.5)));
	double tmp;
	if (f <= 225.0) {
		tmp = Math.log(((-1.0 / t_0) + (((2.0 * (1.0 / Math.PI)) - (f * (0.5 + (Math.PI * (f * -0.041666666666666664))))) / f))) * (-4.0 / Math.PI);
	} else {
		tmp = Math.log(-t_0) * (4.0 / Math.PI);
	}
	return tmp;
}

Python

def code(f):
	t_0 = math.expm1((math.pi * (f * -0.5)))
	tmp = 0
	if f <= 225.0:
		tmp = math.log(((-1.0 / t_0) + (((2.0 * (1.0 / math.pi)) - (f * (0.5 + (math.pi * (f * -0.041666666666666664))))) / f))) * (-4.0 / math.pi)
	else:
		tmp = math.log(-t_0) * (4.0 / math.pi)
	return tmp

Julia

function code(f)
	t_0 = expm1(Float64(pi * Float64(f * -0.5)))
	tmp = 0.0
	if (f <= 225.0)
		tmp = Float64(log(Float64(Float64(-1.0 / t_0) + Float64(Float64(Float64(2.0 * Float64(1.0 / pi)) - Float64(f * Float64(0.5 + Float64(pi * Float64(f * -0.041666666666666664))))) / f))) * Float64(-4.0 / pi));
	else
		tmp = Float64(log(Float64(-t_0)) * Float64(4.0 / pi));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[(Exp[N[(Pi * N[(f * -0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]}, If[LessEqual[f, 225.0], N[(N[Log[N[(N[(-1.0 / t$95$0), $MachinePrecision] + N[(N[(N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] - N[(f * N[(0.5 + N[(Pi * N[(f * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision], N[(N[Log[(-t$95$0)], $MachinePrecision] * N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if f < 225

   1. Initial program 5.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in f around 0 98.8%

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

      1. distribute-lft-in 98.8%

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

      2. *-commutative 98.8%

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

      3. *-commutative 98.8%

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

   7. Applied egg-rr 98.8%

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

      1. associate-*r* 98.8%

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

      2. associate-*r* 98.8%

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

      3. distribute-lft-out 98.8%

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

      4. metadata-eval 98.8%

         \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \left(\left(f \cdot \pi\right) \cdot \color{blue}{-0.041666666666666664}\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

      5. *-commutative 98.8%

         \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \left(\color{blue}{\left(\pi \cdot f\right)} \cdot -0.041666666666666664\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

      6. associate-*l* 98.8%

         \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \color{blue}{\left(\pi \cdot \left(f \cdot -0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   9. Simplified 98.8%

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

   if 225 < f

   1. Initial program 0.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in f around 0 3.2%

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

      1. *-commutative 3.2%

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

      2. associate-/r* 3.2%

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

   7. Simplified 3.2%

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

   8. Taylor expanded in f around inf 100.0%

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

      1. distribute-neg-frac 100.0%

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

      2. metadata-eval 100.0%

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

      3. expm1-define 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in f around inf 100.0%

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

       1. associate-*r/ 100.0%

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

       2. expm1-define 100.0%

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

       3. *-commutative 100.0%

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

       4. *-commutative 100.0%

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

       5. associate-*r* 100.0%

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

       6. *-commutative 100.0%

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

       7. remove-double-neg 100.0%

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

       8. distribute-neg-frac 100.0%

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

       9. distribute-frac-neg2 100.0%

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

       10. distribute-lft-neg-in 100.0%

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

       11. associate-/l* 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 98.8%

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

Alternative 3: 98.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \frac{1}{\pi}\\ \mathbf{if}\;f \leq 225:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{t\_0 - f \cdot \left(0.5 + \pi \cdot \left(f \cdot -0.041666666666666664\right)\right)}{f} + \frac{t\_0 + f \cdot \left(0.5 - f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)\right) \cdot \frac{4}{\pi}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* 2.0 (/ 1.0 PI))))
   (if (<= f 225.0)
     (*
      (/ -4.0 PI)
      (log
       (+
        (/ (- t_0 (* f (+ 0.5 (* PI (* f -0.041666666666666664))))) f)
        (/
         (+
          t_0
          (* f (- 0.5 (* f (+ (* PI -0.125) (* PI 0.08333333333333333))))))
         f))))
     (* (log (- (expm1 (* PI (* f -0.5))))) (/ 4.0 PI)))))

C

double code(double f) {
	double t_0 = 2.0 * (1.0 / ((double) M_PI));
	double tmp;
	if (f <= 225.0) {
		tmp = (-4.0 / ((double) M_PI)) * log((((t_0 - (f * (0.5 + (((double) M_PI) * (f * -0.041666666666666664))))) / f) + ((t_0 + (f * (0.5 - (f * ((((double) M_PI) * -0.125) + (((double) M_PI) * 0.08333333333333333)))))) / f)));
	} else {
		tmp = log(-expm1((((double) M_PI) * (f * -0.5)))) * (4.0 / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = 2.0 * (1.0 / Math.PI);
	double tmp;
	if (f <= 225.0) {
		tmp = (-4.0 / Math.PI) * Math.log((((t_0 - (f * (0.5 + (Math.PI * (f * -0.041666666666666664))))) / f) + ((t_0 + (f * (0.5 - (f * ((Math.PI * -0.125) + (Math.PI * 0.08333333333333333)))))) / f)));
	} else {
		tmp = Math.log(-Math.expm1((Math.PI * (f * -0.5)))) * (4.0 / Math.PI);
	}
	return tmp;
}

Python

def code(f):
	t_0 = 2.0 * (1.0 / math.pi)
	tmp = 0
	if f <= 225.0:
		tmp = (-4.0 / math.pi) * math.log((((t_0 - (f * (0.5 + (math.pi * (f * -0.041666666666666664))))) / f) + ((t_0 + (f * (0.5 - (f * ((math.pi * -0.125) + (math.pi * 0.08333333333333333)))))) / f)))
	else:
		tmp = math.log(-math.expm1((math.pi * (f * -0.5)))) * (4.0 / math.pi)
	return tmp

Julia

function code(f)
	t_0 = Float64(2.0 * Float64(1.0 / pi))
	tmp = 0.0
	if (f <= 225.0)
		tmp = Float64(Float64(-4.0 / pi) * log(Float64(Float64(Float64(t_0 - Float64(f * Float64(0.5 + Float64(pi * Float64(f * -0.041666666666666664))))) / f) + Float64(Float64(t_0 + Float64(f * Float64(0.5 - Float64(f * Float64(Float64(pi * -0.125) + Float64(pi * 0.08333333333333333)))))) / f))));
	else
		tmp = Float64(log(Float64(-expm1(Float64(pi * Float64(f * -0.5))))) * Float64(4.0 / pi));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[f, 225.0], N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(N[(N[(t$95$0 - N[(f * N[(0.5 + N[(Pi * N[(f * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision] + N[(N[(t$95$0 + N[(f * N[(0.5 - N[(f * N[(N[(Pi * -0.125), $MachinePrecision] + N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[(-N[(Exp[N[(Pi * N[(f * -0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision])], $MachinePrecision] * N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if f < 225

   1. Initial program 5.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in f around 0 98.8%

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

   6. Taylor expanded in f around 0 98.8%

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

      1. distribute-lft-in 98.8%

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

      2. *-commutative 98.8%

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

      3. *-commutative 98.8%

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

   8. Applied egg-rr 98.8%

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

      1. associate-*r* 98.8%

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

      2. associate-*r* 98.8%

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

      3. distribute-lft-out 98.8%

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

      4. metadata-eval 98.8%

         \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \left(\left(f \cdot \pi\right) \cdot \color{blue}{-0.041666666666666664}\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

      5. *-commutative 98.8%

         \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \left(\color{blue}{\left(\pi \cdot f\right)} \cdot -0.041666666666666664\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

      6. associate-*l* 98.8%

         \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \color{blue}{\left(\pi \cdot \left(f \cdot -0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   10. Simplified 98.8%

       \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \color{blue}{\left(\pi \cdot \left(f \cdot -0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   if 225 < f

   1. Initial program 0.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in f around 0 3.2%

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

      1. *-commutative 3.2%

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

      2. associate-/r* 3.2%

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

   7. Simplified 3.2%

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

   8. Taylor expanded in f around inf 100.0%

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

      1. distribute-neg-frac 100.0%

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

      2. metadata-eval 100.0%

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

      3. expm1-define 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in f around inf 100.0%

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

       1. associate-*r/ 100.0%

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

       2. expm1-define 100.0%

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

       3. *-commutative 100.0%

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

       4. *-commutative 100.0%

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

       5. associate-*r* 100.0%

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

       6. *-commutative 100.0%

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

       7. remove-double-neg 100.0%

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

       8. distribute-neg-frac 100.0%

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

       9. distribute-frac-neg2 100.0%

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

       10. distribute-lft-neg-in 100.0%

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

       11. associate-/l* 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + \pi \cdot \left(f \cdot -0.041666666666666664\right)\right)}{f} + \frac{2 \cdot \frac{1}{\pi} + f \cdot \left(0.5 - f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)}{f}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)\right) \cdot \frac{4}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \frac{1}{\pi}\\ \frac{-4}{\pi} \cdot \log \left(\frac{t\_0 - f \cdot \left(0.5 + \pi \cdot \left(f \cdot -0.041666666666666664\right)\right)}{f} + \frac{t\_0 + f \cdot \left(0.5 - f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)}{f}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* 2.0 (/ 1.0 PI))))
   (*
    (/ -4.0 PI)
    (log
     (+
      (/ (- t_0 (* f (+ 0.5 (* PI (* f -0.041666666666666664))))) f)
      (/
       (+ t_0 (* f (- 0.5 (* f (+ (* PI -0.125) (* PI 0.08333333333333333))))))
       f))))))

C

double code(double f) {
	double t_0 = 2.0 * (1.0 / ((double) M_PI));
	return (-4.0 / ((double) M_PI)) * log((((t_0 - (f * (0.5 + (((double) M_PI) * (f * -0.041666666666666664))))) / f) + ((t_0 + (f * (0.5 - (f * ((((double) M_PI) * -0.125) + (((double) M_PI) * 0.08333333333333333)))))) / f)));
}

Java

public static double code(double f) {
	double t_0 = 2.0 * (1.0 / Math.PI);
	return (-4.0 / Math.PI) * Math.log((((t_0 - (f * (0.5 + (Math.PI * (f * -0.041666666666666664))))) / f) + ((t_0 + (f * (0.5 - (f * ((Math.PI * -0.125) + (Math.PI * 0.08333333333333333)))))) / f)));
}

Python

def code(f):
	t_0 = 2.0 * (1.0 / math.pi)
	return (-4.0 / math.pi) * math.log((((t_0 - (f * (0.5 + (math.pi * (f * -0.041666666666666664))))) / f) + ((t_0 + (f * (0.5 - (f * ((math.pi * -0.125) + (math.pi * 0.08333333333333333)))))) / f)))

Julia

function code(f)
	t_0 = Float64(2.0 * Float64(1.0 / pi))
	return Float64(Float64(-4.0 / pi) * log(Float64(Float64(Float64(t_0 - Float64(f * Float64(0.5 + Float64(pi * Float64(f * -0.041666666666666664))))) / f) + Float64(Float64(t_0 + Float64(f * Float64(0.5 - Float64(f * Float64(Float64(pi * -0.125) + Float64(pi * 0.08333333333333333)))))) / f))))
end

MATLAB

function tmp = code(f)
	t_0 = 2.0 * (1.0 / pi);
	tmp = (-4.0 / pi) * log((((t_0 - (f * (0.5 + (pi * (f * -0.041666666666666664))))) / f) + ((t_0 + (f * (0.5 - (f * ((pi * -0.125) + (pi * 0.08333333333333333)))))) / f)));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(N[(N[(t$95$0 - N[(f * N[(0.5 + N[(Pi * N[(f * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision] + N[(N[(t$95$0 + N[(f * N[(0.5 - N[(f * N[(N[(Pi * -0.125), $MachinePrecision] + N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

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 98.9%

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

5. Taylor expanded in f around 0 96.2%

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

6. Taylor expanded in f around 0 96.2%

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

   1. distribute-lft-in 96.2%

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

   2. *-commutative 96.2%

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

   3. *-commutative 96.2%

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

8. Applied egg-rr 96.2%

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

   1. associate-*r* 96.2%

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

   2. associate-*r* 96.2%

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

   3. distribute-lft-out 96.2%

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

   4. metadata-eval 96.2%

      \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \left(\left(f \cdot \pi\right) \cdot \color{blue}{-0.041666666666666664}\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   5. *-commutative 96.2%

      \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \left(\color{blue}{\left(\pi \cdot f\right)} \cdot -0.041666666666666664\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   6. associate-*l* 96.2%

      \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \color{blue}{\left(\pi \cdot \left(f \cdot -0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

10. Simplified 96.2%

    \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \color{blue}{\left(\pi \cdot \left(f \cdot -0.041666666666666664\right)\right)} - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

11. Final simplification 96.2%

    \[\leadsto \frac{-4}{\pi} \cdot \log \left(\frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + \pi \cdot \left(f \cdot -0.041666666666666664\right)\right)}{f} + \frac{2 \cdot \frac{1}{\pi} + f \cdot \left(0.5 - f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)}{f}\right) \]
12. Add Preprocessing

Alternative 5: 96.2% 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 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 98.9%

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

5. Taylor expanded in f around 0 96.0%

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

   1. mul-1-neg 96.0%

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

   2. unsub-neg 96.0%

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

7. Simplified 96.0%

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

   1. diff-log 96.0%

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

9. Applied egg-rr 96.0%

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

10. Final simplification 96.0%

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

Reproduce

herbie shell --seed 2024074 
(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)))))))))