VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 99.1%

Time: 9.5s

Alternatives: 8

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1319216836.084853

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: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(f \cdot \pi\right) \cdot -0.25}\\ t_1 := \left(f \cdot \pi\right) \cdot 0.25\\ \mathbf{if}\;f \leq 24:\\ \;\;\;\;\frac{\log 2 + \log \left(\frac{\cosh t\_1}{\sinh t\_1 \cdot 2}\right)}{\pi} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{\pi} \cdot \left(-\log \left(\frac{t\_0 + 1}{1 - t\_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (* (* f PI) -0.25))) (t_1 (* (* f PI) 0.25)))
   (if (<= f 24.0)
     (* (/ (+ (log 2.0) (log (/ (cosh t_1) (* (sinh t_1) 2.0)))) PI) -4.0)
     (* (/ 4.0 PI) (- (log (/ (+ t_0 1.0) (- 1.0 t_0))))))))

C

double code(double f) {
	double t_0 = exp(((f * ((double) M_PI)) * -0.25));
	double t_1 = (f * ((double) M_PI)) * 0.25;
	double tmp;
	if (f <= 24.0) {
		tmp = ((log(2.0) + log((cosh(t_1) / (sinh(t_1) * 2.0)))) / ((double) M_PI)) * -4.0;
	} else {
		tmp = (4.0 / ((double) M_PI)) * -log(((t_0 + 1.0) / (1.0 - t_0)));
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = Math.exp(((f * Math.PI) * -0.25));
	double t_1 = (f * Math.PI) * 0.25;
	double tmp;
	if (f <= 24.0) {
		tmp = ((Math.log(2.0) + Math.log((Math.cosh(t_1) / (Math.sinh(t_1) * 2.0)))) / Math.PI) * -4.0;
	} else {
		tmp = (4.0 / Math.PI) * -Math.log(((t_0 + 1.0) / (1.0 - t_0)));
	}
	return tmp;
}

Python

def code(f):
	t_0 = math.exp(((f * math.pi) * -0.25))
	t_1 = (f * math.pi) * 0.25
	tmp = 0
	if f <= 24.0:
		tmp = ((math.log(2.0) + math.log((math.cosh(t_1) / (math.sinh(t_1) * 2.0)))) / math.pi) * -4.0
	else:
		tmp = (4.0 / math.pi) * -math.log(((t_0 + 1.0) / (1.0 - t_0)))
	return tmp

Julia

function code(f)
	t_0 = exp(Float64(Float64(f * pi) * -0.25))
	t_1 = Float64(Float64(f * pi) * 0.25)
	tmp = 0.0
	if (f <= 24.0)
		tmp = Float64(Float64(Float64(log(2.0) + log(Float64(cosh(t_1) / Float64(sinh(t_1) * 2.0)))) / pi) * -4.0);
	else
		tmp = Float64(Float64(4.0 / pi) * Float64(-log(Float64(Float64(t_0 + 1.0) / Float64(1.0 - t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	t_0 = exp(((f * pi) * -0.25));
	t_1 = (f * pi) * 0.25;
	tmp = 0.0;
	if (f <= 24.0)
		tmp = ((log(2.0) + log((cosh(t_1) / (sinh(t_1) * 2.0)))) / pi) * -4.0;
	else
		tmp = (4.0 / pi) * -log(((t_0 + 1.0) / (1.0 - t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := Block[{t$95$0 = N[Exp[N[(N[(f * Pi), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]}, If[LessEqual[f, 24.0], N[(N[(N[(N[Log[2.0], $MachinePrecision] + N[Log[N[(N[Cosh[t$95$1], $MachinePrecision] / N[(N[Sinh[t$95$1], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(4.0 / Pi), $MachinePrecision] * (-N[Log[N[(N[(t$95$0 + 1.0), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if f < 24

   1. Initial program 6.6%

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

   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 96.9%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot f\right) \cdot -0.25\right)}{2 \cdot \sinh \left(\left(\pi \cdot f\right) \cdot 0.25\right)}\right)}{\pi} \cdot -4} \]
   5. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cosh.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

   6. Applied rewrites 96.8%

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

   if 24 < f

   1. Initial program 6.6%

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

   2. Taylor expanded in f around 0

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

   4. Applied rewrites 5.3%

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

   5. Taylor expanded in f around 0

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

   7. Applied rewrites 6.5%

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

   8. Taylor expanded in f around inf

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

      1. distribute-lft-neg-in N/A

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

      2. metadata-eval N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 6.5

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

   10. Applied rewrites 6.5%

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

   11. Taylor expanded in f around inf

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

       1. distribute-lft-neg-in N/A

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

       2. metadata-eval N/A

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

       3. lower-exp.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-PI.f64 6.5

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

   13. Applied rewrites 6.5%

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

       1. lift-neg.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. distribute-rgt-neg-in N/A

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

   15. Applied rewrites 6.5%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*l/ N/A

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

       5. metadata-eval N/A

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

       6. lower-/.f64 N/A

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

       7. lift-PI.f64 6.5

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

   17. Applied rewrites 6.5%

       \[\leadsto \color{blue}{\frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\left(f \cdot \pi\right) \cdot -0.25} + 1}{1 - e^{\left(f \cdot \pi\right) \cdot -0.25}}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(f \cdot \pi\right) \cdot -0.25}\\ t_1 := \left(f \cdot \pi\right) \cdot 0.25\\ \mathbf{if}\;f \leq 24:\\ \;\;\;\;\frac{\log \cosh t\_1 - \log \sinh t\_1}{\pi} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{\pi} \cdot \left(-\log \left(\frac{t\_0 + 1}{1 - t\_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (* (* f PI) -0.25))) (t_1 (* (* f PI) 0.25)))
   (if (<= f 24.0)
     (* (/ (- (log (cosh t_1)) (log (sinh t_1))) PI) -4.0)
     (* (/ 4.0 PI) (- (log (/ (+ t_0 1.0) (- 1.0 t_0))))))))

C

double code(double f) {
	double t_0 = exp(((f * ((double) M_PI)) * -0.25));
	double t_1 = (f * ((double) M_PI)) * 0.25;
	double tmp;
	if (f <= 24.0) {
		tmp = ((log(cosh(t_1)) - log(sinh(t_1))) / ((double) M_PI)) * -4.0;
	} else {
		tmp = (4.0 / ((double) M_PI)) * -log(((t_0 + 1.0) / (1.0 - t_0)));
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = Math.exp(((f * Math.PI) * -0.25));
	double t_1 = (f * Math.PI) * 0.25;
	double tmp;
	if (f <= 24.0) {
		tmp = ((Math.log(Math.cosh(t_1)) - Math.log(Math.sinh(t_1))) / Math.PI) * -4.0;
	} else {
		tmp = (4.0 / Math.PI) * -Math.log(((t_0 + 1.0) / (1.0 - t_0)));
	}
	return tmp;
}

Python

def code(f):
	t_0 = math.exp(((f * math.pi) * -0.25))
	t_1 = (f * math.pi) * 0.25
	tmp = 0
	if f <= 24.0:
		tmp = ((math.log(math.cosh(t_1)) - math.log(math.sinh(t_1))) / math.pi) * -4.0
	else:
		tmp = (4.0 / math.pi) * -math.log(((t_0 + 1.0) / (1.0 - t_0)))
	return tmp

Julia

function code(f)
	t_0 = exp(Float64(Float64(f * pi) * -0.25))
	t_1 = Float64(Float64(f * pi) * 0.25)
	tmp = 0.0
	if (f <= 24.0)
		tmp = Float64(Float64(Float64(log(cosh(t_1)) - log(sinh(t_1))) / pi) * -4.0);
	else
		tmp = Float64(Float64(4.0 / pi) * Float64(-log(Float64(Float64(t_0 + 1.0) / Float64(1.0 - t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	t_0 = exp(((f * pi) * -0.25));
	t_1 = (f * pi) * 0.25;
	tmp = 0.0;
	if (f <= 24.0)
		tmp = ((log(cosh(t_1)) - log(sinh(t_1))) / pi) * -4.0;
	else
		tmp = (4.0 / pi) * -log(((t_0 + 1.0) / (1.0 - t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := Block[{t$95$0 = N[Exp[N[(N[(f * Pi), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]}, If[LessEqual[f, 24.0], N[(N[(N[(N[Log[N[Cosh[t$95$1], $MachinePrecision]], $MachinePrecision] - N[Log[N[Sinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(4.0 / Pi), $MachinePrecision] * (-N[Log[N[(N[(t$95$0 + 1.0), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if f < 24

   1. Initial program 6.6%

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

   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 96.9%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot f\right) \cdot -0.25\right)}{2 \cdot \sinh \left(\left(\pi \cdot f\right) \cdot 0.25\right)}\right)}{\pi} \cdot -4} \]
   5. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cosh.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

   6. Applied rewrites 96.9%

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

   if 24 < f

   1. Initial program 6.6%

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

   2. Taylor expanded in f around 0

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

   4. Applied rewrites 5.3%

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

   5. Taylor expanded in f around 0

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

   7. Applied rewrites 6.5%

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

   8. Taylor expanded in f around inf

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

      1. distribute-lft-neg-in N/A

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

      2. metadata-eval N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 6.5

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

   10. Applied rewrites 6.5%

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

   11. Taylor expanded in f around inf

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

       1. distribute-lft-neg-in N/A

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

       2. metadata-eval N/A

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

       3. lower-exp.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-PI.f64 6.5

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

   13. Applied rewrites 6.5%

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

       1. lift-neg.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. distribute-rgt-neg-in N/A

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

   15. Applied rewrites 6.5%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*l/ N/A

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

       5. metadata-eval N/A

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

       6. lower-/.f64 N/A

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

       7. lift-PI.f64 6.5

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

   17. Applied rewrites 6.5%

       \[\leadsto \color{blue}{\frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\left(f \cdot \pi\right) \cdot -0.25} + 1}{1 - e^{\left(f \cdot \pi\right) \cdot -0.25}}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(f \cdot \pi\right) \cdot -0.25}\\ t_1 := \left(f \cdot \pi\right) \cdot 0.25\\ \mathbf{if}\;f \leq 24:\\ \;\;\;\;\frac{\log \left(\frac{\cosh t\_1}{\sinh t\_1}\right)}{\pi} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{\pi} \cdot \left(-\log \left(\frac{t\_0 + 1}{1 - t\_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (* (* f PI) -0.25))) (t_1 (* (* f PI) 0.25)))
   (if (<= f 24.0)
     (* (/ (log (/ (cosh t_1) (sinh t_1))) PI) -4.0)
     (* (/ 4.0 PI) (- (log (/ (+ t_0 1.0) (- 1.0 t_0))))))))

C

double code(double f) {
	double t_0 = exp(((f * ((double) M_PI)) * -0.25));
	double t_1 = (f * ((double) M_PI)) * 0.25;
	double tmp;
	if (f <= 24.0) {
		tmp = (log((cosh(t_1) / sinh(t_1))) / ((double) M_PI)) * -4.0;
	} else {
		tmp = (4.0 / ((double) M_PI)) * -log(((t_0 + 1.0) / (1.0 - t_0)));
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = Math.exp(((f * Math.PI) * -0.25));
	double t_1 = (f * Math.PI) * 0.25;
	double tmp;
	if (f <= 24.0) {
		tmp = (Math.log((Math.cosh(t_1) / Math.sinh(t_1))) / Math.PI) * -4.0;
	} else {
		tmp = (4.0 / Math.PI) * -Math.log(((t_0 + 1.0) / (1.0 - t_0)));
	}
	return tmp;
}

Python

def code(f):
	t_0 = math.exp(((f * math.pi) * -0.25))
	t_1 = (f * math.pi) * 0.25
	tmp = 0
	if f <= 24.0:
		tmp = (math.log((math.cosh(t_1) / math.sinh(t_1))) / math.pi) * -4.0
	else:
		tmp = (4.0 / math.pi) * -math.log(((t_0 + 1.0) / (1.0 - t_0)))
	return tmp

Julia

function code(f)
	t_0 = exp(Float64(Float64(f * pi) * -0.25))
	t_1 = Float64(Float64(f * pi) * 0.25)
	tmp = 0.0
	if (f <= 24.0)
		tmp = Float64(Float64(log(Float64(cosh(t_1) / sinh(t_1))) / pi) * -4.0);
	else
		tmp = Float64(Float64(4.0 / pi) * Float64(-log(Float64(Float64(t_0 + 1.0) / Float64(1.0 - t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	t_0 = exp(((f * pi) * -0.25));
	t_1 = (f * pi) * 0.25;
	tmp = 0.0;
	if (f <= 24.0)
		tmp = (log((cosh(t_1) / sinh(t_1))) / pi) * -4.0;
	else
		tmp = (4.0 / pi) * -log(((t_0 + 1.0) / (1.0 - t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := Block[{t$95$0 = N[Exp[N[(N[(f * Pi), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]}, If[LessEqual[f, 24.0], N[(N[(N[Log[N[(N[Cosh[t$95$1], $MachinePrecision] / N[Sinh[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(4.0 / Pi), $MachinePrecision] * (-N[Log[N[(N[(t$95$0 + 1.0), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if f < 24

   1. Initial program 6.6%

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

   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 96.9%

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

   6. Applied rewrites 96.9%

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

   if 24 < f

   1. Initial program 6.6%

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

   2. Taylor expanded in f around 0

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

   4. Applied rewrites 5.3%

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

   5. Taylor expanded in f around 0

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

   7. Applied rewrites 6.5%

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

   8. Taylor expanded in f around inf

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

      1. distribute-lft-neg-in N/A

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

      2. metadata-eval N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 6.5

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

   10. Applied rewrites 6.5%

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

   11. Taylor expanded in f around inf

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

       1. distribute-lft-neg-in N/A

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

       2. metadata-eval N/A

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

       3. lower-exp.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-PI.f64 6.5

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

   13. Applied rewrites 6.5%

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

       1. lift-neg.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. distribute-rgt-neg-in N/A

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

   15. Applied rewrites 6.5%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*l/ N/A

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

       5. metadata-eval N/A

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

       6. lower-/.f64 N/A

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

       7. lift-PI.f64 6.5

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

   17. Applied rewrites 6.5%

       \[\leadsto \color{blue}{\frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\left(f \cdot \pi\right) \cdot -0.25} + 1}{1 - e^{\left(f \cdot \pi\right) \cdot -0.25}}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.9% 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 6.6%

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

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 96.9%

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

6. Applied rewrites 96.9%

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

Alternative 5: 96.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{\left(0.03125 \cdot \left(f \cdot f\right)\right) \cdot \left(\pi \cdot \pi\right) - \log \sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}{\pi} \cdot -4 \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ (- (* (* 0.03125 (* f f)) (* PI PI)) (log (sinh (* (* f PI) 0.25)))) PI)
  -4.0))

C

double code(double f) {
	return ((((0.03125 * (f * f)) * (((double) M_PI) * ((double) M_PI))) - log(sinh(((f * ((double) M_PI)) * 0.25)))) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return ((((0.03125 * (f * f)) * (Math.PI * Math.PI)) - Math.log(Math.sinh(((f * Math.PI) * 0.25)))) / Math.PI) * -4.0;
}

Python

def code(f):
	return ((((0.03125 * (f * f)) * (math.pi * math.pi)) - math.log(math.sinh(((f * math.pi) * 0.25)))) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(Float64(Float64(Float64(0.03125 * Float64(f * f)) * Float64(pi * pi)) - log(sinh(Float64(Float64(f * pi) * 0.25)))) / pi) * -4.0)
end

MATLAB

function tmp = code(f)
	tmp = ((((0.03125 * (f * f)) * (pi * pi)) - log(sinh(((f * pi) * 0.25)))) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[(N[(N[(0.03125 * N[(f * f), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] - N[Log[N[Sinh[N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 6.6%

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

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 96.9%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot f\right) \cdot -0.25\right)}{2 \cdot \sinh \left(\left(\pi \cdot f\right) \cdot 0.25\right)}\right)}{\pi} \cdot -4} \]
5. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cosh.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-sinh.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/r* N/A

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

6. Applied rewrites 96.9%

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

7. Taylor expanded in f around 0

   \[\leadsto \frac{\frac{1}{32} \cdot \left({f}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) - \log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi} \cdot -4 \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\left(\frac{1}{32} \cdot {f}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} - \log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi} \cdot -4 \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{32} \cdot {f}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} - \log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi} \cdot -4 \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{32} \cdot {f}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} - \log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi} \cdot -4 \]

   4. unpow2 N/A

      \[\leadsto \frac{\left(\frac{1}{32} \cdot \left(f \cdot f\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2} - \log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi} \cdot -4 \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{32} \cdot \left(f \cdot f\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2} - \log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi} \cdot -4 \]

   6. unpow2 N/A

      \[\leadsto \frac{\left(\frac{1}{32} \cdot \left(f \cdot f\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) - \log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi} \cdot -4 \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{32} \cdot \left(f \cdot f\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) - \log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi} \cdot -4 \]

   8. lift-PI.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{32} \cdot \left(f \cdot f\right)\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right) - \log \sinh \left(\left(f \cdot \pi\right) \cdot \frac{1}{4}\right)}{\pi} \cdot -4 \]

   9. lift-PI.f64 96.2

      \[\leadsto \frac{\left(0.03125 \cdot \left(f \cdot f\right)\right) \cdot \left(\pi \cdot \pi\right) - \log \sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}{\pi} \cdot -4 \]

9. Applied rewrites 96.2%

   \[\leadsto \frac{\left(0.03125 \cdot \left(f \cdot f\right)\right) \cdot \left(\pi \cdot \pi\right) - \log \sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}{\pi} \cdot -4 \]
10. Add Preprocessing

Alternative 6: 95.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot f\right) \cdot -0.25\right)}{\left(0.5 \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (/ (log (/ (* 2.0 (cosh (* (* PI f) -0.25))) (* (* 0.5 PI) f))) PI) -4.0))

C

double code(double f) {
	return (log(((2.0 * cosh(((((double) M_PI) * f) * -0.25))) / ((0.5 * ((double) M_PI)) * f))) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return (Math.log(((2.0 * Math.cosh(((Math.PI * f) * -0.25))) / ((0.5 * Math.PI) * f))) / Math.PI) * -4.0;
}

Python

def code(f):
	return (math.log(((2.0 * math.cosh(((math.pi * f) * -0.25))) / ((0.5 * math.pi) * f))) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(log(Float64(Float64(2.0 * cosh(Float64(Float64(pi * f) * -0.25))) / Float64(Float64(0.5 * pi) * f))) / pi) * -4.0)
end

MATLAB

function tmp = code(f)
	tmp = (log(((2.0 * cosh(((pi * f) * -0.25))) / ((0.5 * pi) * f))) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[Log[N[(N[(2.0 * N[Cosh[N[(N[(Pi * f), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * Pi), $MachinePrecision] * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 6.6%

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

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 96.9%

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

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. distribute-rgt-out-- N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 95.8

      \[\leadsto \frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot f\right) \cdot -0.25\right)}{\left(0.5 \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]

7. Applied rewrites 95.8%

   \[\leadsto \frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot f\right) \cdot -0.25\right)}{\left(0.5 \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]
8. Add Preprocessing

Alternative 7: 95.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \frac{\log 2 - \log \left(\left(0.5 \cdot \pi\right) \cdot f\right)}{\pi} \cdot -4 \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (/ (- (log 2.0) (log (* (* 0.5 PI) f))) PI) -4.0))

C

double code(double f) {
	return ((log(2.0) - log(((0.5 * ((double) M_PI)) * f))) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return ((Math.log(2.0) - Math.log(((0.5 * Math.PI) * f))) / Math.PI) * -4.0;
}

Python

def code(f):
	return ((math.log(2.0) - math.log(((0.5 * math.pi) * f))) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(Float64(log(2.0) - log(Float64(Float64(0.5 * pi) * f))) / pi) * -4.0)
end

MATLAB

function tmp = code(f)
	tmp = ((log(2.0) - log(((0.5 * pi) * f))) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[(N[Log[2.0], $MachinePrecision] - N[Log[N[(N[(0.5 * Pi), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 6.6%

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

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. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. log-div N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. distribute-rgt-out-- N/A

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

   9. *-commutative N/A

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

   10. sum-log N/A

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

   11. lower--.f64 N/A

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

   12. lower-log.f64 N/A

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

   13. sum-log N/A

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

   14. lower-log.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

6. Applied rewrites 95.8%

   \[\leadsto \frac{\log 2 - \log \left(\left(0.5 \cdot \pi\right) \cdot f\right)}{\pi} \cdot -4 \]
7. Add Preprocessing

Alternative 8: 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 6.6%

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

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 2025131 
(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)))))))))