VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 98.9%

Time: 9.0s

Alternatives: 5

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1500047868.292262

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := \log \left(\cosh t\_0 \cdot 2\right)\\ t_2 := \log \left(\sinh t\_0 \cdot 2\right)\\ t_3 := \left(f \cdot \pi\right) \cdot 0.25\\ \mathbf{if}\;f \leq 920:\\ \;\;\;\;\frac{\log \left(\frac{\cosh t\_3}{\sinh t\_3}\right)}{\pi} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;-\frac{1}{\frac{\pi}{4}} \cdot \frac{{\log 2}^{3} - {\log \left(\left(\pi \cdot f\right) \cdot 0.5\right)}^{3}}{\mathsf{fma}\left(t\_1, t\_1, \mathsf{fma}\left(t\_2, t\_2, t\_1 \cdot t\_2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f))
        (t_1 (log (* (cosh t_0) 2.0)))
        (t_2 (log (* (sinh t_0) 2.0)))
        (t_3 (* (* f PI) 0.25)))
   (if (<= f 920.0)
     (* (/ (log (/ (cosh t_3) (sinh t_3))) PI) -4.0)
     (-
      (*
       (/ 1.0 (/ PI 4.0))
       (/
        (- (pow (log 2.0) 3.0) (pow (log (* (* PI f) 0.5)) 3.0))
        (fma t_1 t_1 (fma t_2 t_2 (* t_1 t_2)))))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = log((cosh(t_0) * 2.0));
	double t_2 = log((sinh(t_0) * 2.0));
	double t_3 = (f * ((double) M_PI)) * 0.25;
	double tmp;
	if (f <= 920.0) {
		tmp = (log((cosh(t_3) / sinh(t_3))) / ((double) M_PI)) * -4.0;
	} else {
		tmp = -((1.0 / (((double) M_PI) / 4.0)) * ((pow(log(2.0), 3.0) - pow(log(((((double) M_PI) * f) * 0.5)), 3.0)) / fma(t_1, t_1, fma(t_2, t_2, (t_1 * t_2)))));
	}
	return tmp;
}

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = log(Float64(cosh(t_0) * 2.0))
	t_2 = log(Float64(sinh(t_0) * 2.0))
	t_3 = Float64(Float64(f * pi) * 0.25)
	tmp = 0.0
	if (f <= 920.0)
		tmp = Float64(Float64(log(Float64(cosh(t_3) / sinh(t_3))) / pi) * -4.0);
	else
		tmp = Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * Float64(Float64((log(2.0) ^ 3.0) - (log(Float64(Float64(pi * f) * 0.5)) ^ 3.0)) / fma(t_1, t_1, fma(t_2, t_2, Float64(t_1 * t_2))))));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[(N[Cosh[t$95$0], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Log[N[(N[Sinh[t$95$0], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]}, If[LessEqual[f, 920.0], N[(N[(N[Log[N[(N[Cosh[t$95$3], $MachinePrecision] / N[Sinh[t$95$3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * -4.0), $MachinePrecision], (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Log[2.0], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[N[(N[(Pi * f), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$1 + N[(t$95$2 * t$95$2 + N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]]]

Derivation

1. Split input into 2 regimes

2. if f < 920

   1. Initial program 7.0%

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

   2. Taylor expanded in f around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot f\right) \cdot -0.25\right)}{2 \cdot \sinh \left(\left(\pi \cdot f\right) \cdot 0.25\right)}\right)}{\pi} \cdot -4} \]

   6. Applied rewrites 98.9%

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

   if 920 < f

   1. Initial program 0.0%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

   3. Applied rewrites 0.0%

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

   5. Applied rewrites 0.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\frac{{\log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right)}^{3} - {\log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right)}^{3}}{\mathsf{fma}\left(\log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \mathsf{fma}\left(\log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right) \cdot \log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right)\right)\right)}} \]

   6. Taylor expanded in f around 0

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \frac{\color{blue}{{\log 2}^{3} - {\left(\log f + \log \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}^{3}}}{\mathsf{fma}\left(\log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \mathsf{fma}\left(\log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right) \cdot \log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \frac{{\log 2}^{3} - \color{blue}{{\left(\log f + \log \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}^{3}}}{\mathsf{fma}\left(\log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \mathsf{fma}\left(\log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right) \cdot \log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right)\right)\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \frac{{\log 2}^{3} - {\color{blue}{\left(\log f + \log \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}}^{3}}{\mathsf{fma}\left(\log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \mathsf{fma}\left(\log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right) \cdot \log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right)\right)\right)} \]

      3. lift-log.f64 N/A

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \frac{{\log 2}^{3} - {\left(\color{blue}{\log f} + \log \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}^{3}}{\mathsf{fma}\left(\log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \mathsf{fma}\left(\log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right) \cdot \log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right)\right)\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \frac{{\log 2}^{3} - {\left(\log f + \log \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}^{\color{blue}{3}}}{\mathsf{fma}\left(\log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \mathsf{fma}\left(\log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right) \cdot \log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right)\right)\right)} \]

   8. Applied rewrites 100.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \frac{\color{blue}{{\log 2}^{3} - {\log \left(\left(\pi \cdot f\right) \cdot 0.5\right)}^{3}}}{\mathsf{fma}\left(\log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \mathsf{fma}\left(\log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right), \log \left(\cosh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right) \cdot \log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(f \cdot \pi\right) \cdot 0.25\\ \frac{\log \left(\frac{\cosh t\_0}{\sinh t\_0}\right)}{\pi} \cdot -4 \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (* f PI) 0.25)))
   (* (/ (log (/ (cosh t_0) (sinh t_0))) PI) -4.0)))

C

double code(double f) {
	double t_0 = (f * ((double) M_PI)) * 0.25;
	return (log((cosh(t_0) / sinh(t_0))) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	double t_0 = (f * Math.PI) * 0.25;
	return (Math.log((Math.cosh(t_0) / Math.sinh(t_0))) / Math.PI) * -4.0;
}

Python

def code(f):
	t_0 = (f * math.pi) * 0.25
	return (math.log((math.cosh(t_0) / math.sinh(t_0))) / math.pi) * -4.0

Julia

function code(f)
	t_0 = Float64(Float64(f * pi) * 0.25)
	return Float64(Float64(log(Float64(cosh(t_0) / sinh(t_0))) / pi) * -4.0)
end

MATLAB

function tmp = code(f)
	t_0 = (f * pi) * 0.25;
	tmp = (log((cosh(t_0) / sinh(t_0))) / pi) * -4.0;
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]}, N[(N[(N[Log[N[(N[Cosh[t$95$0], $MachinePrecision] / N[Sinh[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in f around inf

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)}} \]
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 97.0%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot f\right) \cdot -0.25\right)}{2 \cdot \sinh \left(\left(\pi \cdot f\right) \cdot 0.25\right)}\right)}{\pi} \cdot -4} \]

6. Applied rewrites 97.0%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{\cosh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}{\sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}\right)}{\pi} \cdot -4} \]
7. Add Preprocessing

Alternative 3: 96.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ -\frac{4 \cdot \log \left(\frac{\mathsf{fma}\left(\left(0.08333333333333333 \cdot \pi\right) \cdot f, f, \frac{4}{\pi}\right)}{f}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (/
   (* 4.0 (log (/ (fma (* (* 0.08333333333333333 PI) f) f (/ 4.0 PI)) f)))
   PI)))

C

double code(double f) {
	return -((4.0 * log((fma(((0.08333333333333333 * ((double) M_PI)) * f), f, (4.0 / ((double) M_PI))) / f))) / ((double) M_PI));
}

Julia

function code(f)
	return Float64(-Float64(Float64(4.0 * log(Float64(fma(Float64(Float64(0.08333333333333333 * pi) * f), f, Float64(4.0 / pi)) / f))) / pi))
end

Wolfram

code[f_] := (-N[(N[(4.0 * N[Log[N[(N[(N[(N[(0.08333333333333333 * Pi), $MachinePrecision] * f), $MachinePrecision] * f + N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision])

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in f around 0

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{f \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + f \cdot \left(\frac{1}{16} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}}{{\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)\right) + 2 \cdot \frac{1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{f}\right)} \]

4. Applied rewrites 96.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \mathsf{fma}\left(\frac{\pi}{\pi \cdot 0.5}, 0, \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi}, 0.125, -2 \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333}{\left(\pi \cdot \pi\right) \cdot 0.25}\right) \cdot f\right) \cdot f\right)}{f}\right)} \]

5. Taylor expanded in f around 0

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \left(f \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot f\right)}{f}\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \left(\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right) \cdot f\right)}{f}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \left(\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right) \cdot f\right)}{f}\right) \]

   3. distribute-rgt-out N/A

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{24} + \frac{1}{8}\right)\right) \cdot f\right) \cdot f\right)}{f}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{24} + \frac{1}{8}\right)\right) \cdot f\right) \cdot f\right)}{f}\right) \]

   5. lift-PI.f64 N/A

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \left(\left(\pi \cdot \left(\frac{-1}{24} + \frac{1}{8}\right)\right) \cdot f\right) \cdot f\right)}{f}\right) \]

   6. metadata-eval 96.3

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \left(\left(\pi \cdot 0.08333333333333333\right) \cdot f\right) \cdot f\right)}{f}\right) \]

7. Applied rewrites 96.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \left(\left(\pi \cdot 0.08333333333333333\right) \cdot f\right) \cdot f\right)}{f}\right) \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto -\color{blue}{\frac{1}{\frac{\pi}{4}}} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \left(\left(\pi \cdot \frac{1}{12}\right) \cdot f\right) \cdot f\right)}{f}\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto -\frac{1}{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \left(\left(\pi \cdot \frac{1}{12}\right) \cdot f\right) \cdot f\right)}{f}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto -\frac{1}{\color{blue}{\frac{\mathsf{PI}\left(\right)}{4}}} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \left(\left(\pi \cdot \frac{1}{12}\right) \cdot f\right) \cdot f\right)}{f}\right) \]

   4. associate-/r/ N/A

      \[\leadsto -\color{blue}{\left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 4\right)} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \left(\left(\pi \cdot \frac{1}{12}\right) \cdot f\right) \cdot f\right)}{f}\right) \]

   5. *-commutative N/A

      \[\leadsto -\color{blue}{\left(4 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \left(\left(\pi \cdot \frac{1}{12}\right) \cdot f\right) \cdot f\right)}{f}\right) \]

   6. associate-*r/ N/A

      \[\leadsto -\color{blue}{\frac{4 \cdot 1}{\mathsf{PI}\left(\right)}} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \left(\left(\pi \cdot \frac{1}{12}\right) \cdot f\right) \cdot f\right)}{f}\right) \]

   7. metadata-eval N/A

      \[\leadsto -\frac{\color{blue}{4}}{\mathsf{PI}\left(\right)} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \left(\left(\pi \cdot \frac{1}{12}\right) \cdot f\right) \cdot f\right)}{f}\right) \]

   8. lower-/.f64 N/A

      \[\leadsto -\color{blue}{\frac{4}{\mathsf{PI}\left(\right)}} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \left(\left(\pi \cdot \frac{1}{12}\right) \cdot f\right) \cdot f\right)}{f}\right) \]

   9. lift-PI.f64 96.3

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

9. Applied rewrites 96.3%

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

11. Applied rewrites 96.4%

    \[\leadsto -\color{blue}{\frac{4 \cdot \log \left(\frac{\mathsf{fma}\left(\left(0.08333333333333333 \cdot \pi\right) \cdot f, f, \frac{4}{\pi}\right)}{f}\right)}{\pi}} \]
12. Add Preprocessing

Alternative 4: 95.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 95.8%

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

5. Taylor expanded in f around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-PI.f64 95.8

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

7. Applied rewrites 95.8%

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

8. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. diff-log N/A

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

   3. metadata-eval N/A

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

   4. associate-*l/ N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. mul-1-neg N/A

      \[\leadsto \frac{\log \left(\frac{2}{\pi} \cdot 2\right) + \left(\mathsf{neg}\left(\log f\right)\right)}{\pi} \cdot -4 \]

   8. log-rec N/A

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

   9. sum-log N/A

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

   10. lower-log.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. associate-*l/ N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 N/A

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

   17. lift-PI.f64 N/A

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

   18. lower-/.f64 95.8

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

10. Applied rewrites 95.8%

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

    1. lift-log.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-PI.f64 N/A

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

    4. lift-/.f64 N/A

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

    5. lift-/.f64 N/A

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

    6. lift-/.f64 N/A

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

    7. lift-PI.f64 N/A

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

    8. log-prod N/A

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

    9. lift-PI.f64 N/A

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

    10. lift-/.f64 N/A

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

    11. log-rec N/A

        \[\leadsto \frac{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + \left(\mathsf{neg}\left(\log f\right)\right)}{\pi} \cdot -4 \]

    12. mul-1-neg N/A

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

    13. +-commutative N/A

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

    14. lower-+.f64 N/A

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

    15. mul-1-neg N/A

        \[\leadsto \frac{\left(\mathsf{neg}\left(\log f\right)\right) + \log \left(\frac{4}{\mathsf{PI}\left(\right)}\right)}{\pi} \cdot -4 \]

    16. lower-neg.f64 N/A

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

    17. lower-log.f64 N/A

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

    18. lift-/.f64 N/A

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

    19. lift-PI.f64 N/A

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

    20. lower-log.f64 95.9

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

12. Applied rewrites 95.9%

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

Alternative 5: 95.8% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 95.8%

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

5. Taylor expanded in f around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-PI.f64 95.8

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

7. Applied rewrites 95.8%

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

Reproduce

herbie shell --seed 2025112 
(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)))))))))