VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.1%

Time: 6.8s

Alternatives: 16

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1100616206.450487

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{4} \cdot \left(-f\right)}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot f, 0.0026041666666666665, 0.03125 \cdot \left(\pi \cdot \pi\right)\right), f, 0.25 \cdot \pi\right), f, 1\right)\\ \mathbf{if}\;f \leq 23:\\ \;\;\;\;\left(\frac{\log \cosh \left(\left(f \cdot \pi\right) \cdot -0.25\right)}{\pi} - \frac{\log \sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}{\pi}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{\pi} \cdot \left(-\log \left(\frac{t\_0 + t\_1}{t\_1 - t\_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (* (/ PI 4.0) (- f))))
        (t_1
         (fma
          (fma
           (fma
            (* (* (* PI PI) PI) f)
            0.0026041666666666665
            (* 0.03125 (* PI PI)))
           f
           (* 0.25 PI))
          f
          1.0)))
   (if (<= f 23.0)
     (*
      (-
       (/ (log (cosh (* (* f PI) -0.25))) PI)
       (/ (log (sinh (* (* f PI) 0.25))) PI))
      -4.0)
     (* (/ 4.0 PI) (- (log (/ (+ t_0 t_1) (- t_1 t_0))))))))

C

double code(double f) {
	double t_0 = exp(((((double) M_PI) / 4.0) * -f));
	double t_1 = fma(fma(fma((((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * f), 0.0026041666666666665, (0.03125 * (((double) M_PI) * ((double) M_PI)))), f, (0.25 * ((double) M_PI))), f, 1.0);
	double tmp;
	if (f <= 23.0) {
		tmp = ((log(cosh(((f * ((double) M_PI)) * -0.25))) / ((double) M_PI)) - (log(sinh(((f * ((double) M_PI)) * 0.25))) / ((double) M_PI))) * -4.0;
	} else {
		tmp = (4.0 / ((double) M_PI)) * -log(((t_0 + t_1) / (t_1 - t_0)));
	}
	return tmp;
}

Julia

function code(f)
	t_0 = exp(Float64(Float64(pi / 4.0) * Float64(-f)))
	t_1 = fma(fma(fma(Float64(Float64(Float64(pi * pi) * pi) * f), 0.0026041666666666665, Float64(0.03125 * Float64(pi * pi))), f, Float64(0.25 * pi)), f, 1.0)
	tmp = 0.0
	if (f <= 23.0)
		tmp = Float64(Float64(Float64(log(cosh(Float64(Float64(f * pi) * -0.25))) / pi) - Float64(log(sinh(Float64(Float64(f * pi) * 0.25))) / pi)) * -4.0);
	else
		tmp = Float64(Float64(4.0 / pi) * Float64(-log(Float64(Float64(t_0 + t_1) / Float64(t_1 - t_0)))));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * (-f)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * f), $MachinePrecision] * 0.0026041666666666665 + N[(0.03125 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * f + N[(0.25 * Pi), $MachinePrecision]), $MachinePrecision] * f + 1.0), $MachinePrecision]}, If[LessEqual[f, 23.0], N[(N[(N[(N[Log[N[Cosh[N[(N[(f * Pi), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] - N[(N[Log[N[Sinh[N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(4.0 / Pi), $MachinePrecision] * (-N[Log[N[(N[(t$95$0 + t$95$1), $MachinePrecision] / N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if f < 23

   1. Initial program 6.7%

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

   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 99.4%

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

      1. lift-/.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-cosh.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      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-sinh.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-log.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 99.4%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

   8. Applied rewrites 99.4%

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

   if 23 < f

   1. Initial program 3.0%

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

   2. Taylor expanded in f around 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 1.7%

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

   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 86.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 86.1%

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

   10. Taylor expanded in f around 0

       \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \color{blue}{\left(1 + f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) + f \cdot \left(\frac{1}{384} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{32} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}}{1 - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) + f \cdot \left(\frac{1}{384} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{32} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \color{blue}{1}\right)}{1 - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) + f \cdot \left(\frac{1}{384} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{32} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot f + 1\right)}{1 - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) + f \cdot \left(\frac{1}{384} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{32} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{f}, 1\right)}{1 - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

   12. Applied rewrites 3.6%

       \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot f, 0.0026041666666666665, 0.03125 \cdot \left(\pi \cdot \pi\right)\right), f, 0.25 \cdot \pi\right), f, 1\right)}}{1 - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

   13. Taylor expanded in f around 0

       \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot f, \frac{1}{384}, \frac{1}{32} \cdot \left(\pi \cdot \pi\right)\right), f, \frac{1}{4} \cdot \pi\right), f, 1\right)}{\color{blue}{\left(1 + f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) + f \cdot \left(\frac{1}{384} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{32} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]
   14. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot f, \frac{1}{384}, \frac{1}{32} \cdot \left(\pi \cdot \pi\right)\right), f, \frac{1}{4} \cdot \pi\right), f, 1\right)}{\left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) + f \cdot \left(\frac{1}{384} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{32} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \color{blue}{1}\right) - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot f, \frac{1}{384}, \frac{1}{32} \cdot \left(\pi \cdot \pi\right)\right), f, \frac{1}{4} \cdot \pi\right), f, 1\right)}{\left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) + f \cdot \left(\frac{1}{384} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{32} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot f + 1\right) - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot f, \frac{1}{384}, \frac{1}{32} \cdot \left(\pi \cdot \pi\right)\right), f, \frac{1}{4} \cdot \pi\right), f, 1\right)}{\mathsf{fma}\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) + f \cdot \left(\frac{1}{384} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{32} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{f}, 1\right) - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

   15. Applied rewrites 86.1%

       \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot f, 0.0026041666666666665, 0.03125 \cdot \left(\pi \cdot \pi\right)\right), f, 0.25 \cdot \pi\right), f, 1\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot f, 0.0026041666666666665, 0.03125 \cdot \left(\pi \cdot \pi\right)\right), f, 0.25 \cdot \pi\right), f, 1\right)} - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{4} \cdot \left(-f\right)}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, 0.03125, 0.25 \cdot \pi\right), f, 1\right)\\ \mathbf{if}\;f \leq 23:\\ \;\;\;\;\left(\frac{\log \cosh \left(\left(f \cdot \pi\right) \cdot -0.25\right)}{\pi} - \frac{\log \sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}{\pi}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{\pi} \cdot \left(-\log \left(\frac{t\_0 + t\_1}{t\_1 - t\_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (* (/ PI 4.0) (- f))))
        (t_1 (fma (fma (* (* PI PI) f) 0.03125 (* 0.25 PI)) f 1.0)))
   (if (<= f 23.0)
     (*
      (-
       (/ (log (cosh (* (* f PI) -0.25))) PI)
       (/ (log (sinh (* (* f PI) 0.25))) PI))
      -4.0)
     (* (/ 4.0 PI) (- (log (/ (+ t_0 t_1) (- t_1 t_0))))))))

C

double code(double f) {
	double t_0 = exp(((((double) M_PI) / 4.0) * -f));
	double t_1 = fma(fma(((((double) M_PI) * ((double) M_PI)) * f), 0.03125, (0.25 * ((double) M_PI))), f, 1.0);
	double tmp;
	if (f <= 23.0) {
		tmp = ((log(cosh(((f * ((double) M_PI)) * -0.25))) / ((double) M_PI)) - (log(sinh(((f * ((double) M_PI)) * 0.25))) / ((double) M_PI))) * -4.0;
	} else {
		tmp = (4.0 / ((double) M_PI)) * -log(((t_0 + t_1) / (t_1 - t_0)));
	}
	return tmp;
}

Julia

function code(f)
	t_0 = exp(Float64(Float64(pi / 4.0) * Float64(-f)))
	t_1 = fma(fma(Float64(Float64(pi * pi) * f), 0.03125, Float64(0.25 * pi)), f, 1.0)
	tmp = 0.0
	if (f <= 23.0)
		tmp = Float64(Float64(Float64(log(cosh(Float64(Float64(f * pi) * -0.25))) / pi) - Float64(log(sinh(Float64(Float64(f * pi) * 0.25))) / pi)) * -4.0);
	else
		tmp = Float64(Float64(4.0 / pi) * Float64(-log(Float64(Float64(t_0 + t_1) / Float64(t_1 - t_0)))));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * (-f)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * f), $MachinePrecision] * 0.03125 + N[(0.25 * Pi), $MachinePrecision]), $MachinePrecision] * f + 1.0), $MachinePrecision]}, If[LessEqual[f, 23.0], N[(N[(N[(N[Log[N[Cosh[N[(N[(f * Pi), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] - N[(N[Log[N[Sinh[N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(4.0 / Pi), $MachinePrecision] * (-N[Log[N[(N[(t$95$0 + t$95$1), $MachinePrecision] / N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if f < 23

   1. Initial program 6.7%

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

   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 99.4%

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

      1. lift-/.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-cosh.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      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-sinh.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-log.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 99.4%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

   8. Applied rewrites 99.4%

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

   if 23 < f

   1. Initial program 3.0%

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

   2. Taylor expanded in f around 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 1.7%

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

   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 86.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 86.1%

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

   10. Taylor expanded in f around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

          \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\frac{1}{32} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right), \color{blue}{f}, 1\right)}{1 - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

          \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, \frac{1}{32}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right), f, 1\right)}{1 - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, \frac{1}{32}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right), f, 1\right)}{1 - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

       10. lift-PI.f64 N/A

           \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot f, \frac{1}{32}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right), f, 1\right)}{1 - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

       11. lift-PI.f64 N/A

           \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, \frac{1}{32}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right), f, 1\right)}{1 - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, \frac{1}{32}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right), f, 1\right)}{1 - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

       13. lift-PI.f64 3.6

           \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, 0.03125, 0.25 \cdot \pi\right), f, 1\right)}{1 - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

   12. Applied rewrites 3.6%

       \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, 0.03125, 0.25 \cdot \pi\right), f, 1\right)}}{1 - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

   13. Taylor expanded in f around 0

       \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, \frac{1}{32}, \frac{1}{4} \cdot \pi\right), f, 1\right)}{\color{blue}{\left(1 + f \cdot \left(\frac{1}{32} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)} - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]
   14. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, \frac{1}{32}, \frac{1}{4} \cdot \pi\right), f, 1\right)}{\left(f \cdot \left(\frac{1}{32} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{1}\right) - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

          \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, \frac{1}{32}, \frac{1}{4} \cdot \pi\right), f, 1\right)}{\mathsf{fma}\left(\frac{1}{32} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right), \color{blue}{f}, 1\right) - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

          \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, \frac{1}{32}, \frac{1}{4} \cdot \pi\right), f, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, \frac{1}{32}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right), f, 1\right) - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, \frac{1}{32}, \frac{1}{4} \cdot \pi\right), f, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, \frac{1}{32}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right), f, 1\right) - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

       10. lift-PI.f64 N/A

           \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, \frac{1}{32}, \frac{1}{4} \cdot \pi\right), f, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot f, \frac{1}{32}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right), f, 1\right) - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

       11. lift-PI.f64 N/A

           \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, \frac{1}{32}, \frac{1}{4} \cdot \pi\right), f, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, \frac{1}{32}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right), f, 1\right) - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, \frac{1}{32}, \frac{1}{4} \cdot \pi\right), f, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, \frac{1}{32}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right), f, 1\right) - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

       13. lift-PI.f64 86.1

           \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, 0.03125, 0.25 \cdot \pi\right), f, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, 0.03125, 0.25 \cdot \pi\right), f, 1\right) - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]

   15. Applied rewrites 86.1%

       \[\leadsto \frac{4}{\pi} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot \left(-f\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, 0.03125, 0.25 \cdot \pi\right), f, 1\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, 0.03125, 0.25 \cdot \pi\right), f, 1\right)} - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(f \cdot \pi\right) \cdot 0.25\\ t_1 := \mathsf{fma}\left(f \cdot \pi, 0.25, 1\right)\\ t_2 := e^{-t\_0}\\ \mathbf{if}\;f \leq 23:\\ \;\;\;\;\left(\frac{\log \cosh \left(\left(f \cdot \pi\right) \cdot -0.25\right)}{\pi} - \frac{\log \sinh t\_0}{\pi}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(\frac{t\_1 + t\_2}{t\_1 - t\_2}\right)}^{\left(-\frac{1}{\pi} \cdot 4\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (* f PI) 0.25))
        (t_1 (fma (* f PI) 0.25 1.0))
        (t_2 (exp (- t_0))))
   (if (<= f 23.0)
     (*
      (- (/ (log (cosh (* (* f PI) -0.25))) PI) (/ (log (sinh t_0)) PI))
      -4.0)
     (log (pow (/ (+ t_1 t_2) (- t_1 t_2)) (- (* (/ 1.0 PI) 4.0)))))))

C

double code(double f) {
	double t_0 = (f * ((double) M_PI)) * 0.25;
	double t_1 = fma((f * ((double) M_PI)), 0.25, 1.0);
	double t_2 = exp(-t_0);
	double tmp;
	if (f <= 23.0) {
		tmp = ((log(cosh(((f * ((double) M_PI)) * -0.25))) / ((double) M_PI)) - (log(sinh(t_0)) / ((double) M_PI))) * -4.0;
	} else {
		tmp = log(pow(((t_1 + t_2) / (t_1 - t_2)), -((1.0 / ((double) M_PI)) * 4.0)));
	}
	return tmp;
}

Julia

function code(f)
	t_0 = Float64(Float64(f * pi) * 0.25)
	t_1 = fma(Float64(f * pi), 0.25, 1.0)
	t_2 = exp(Float64(-t_0))
	tmp = 0.0
	if (f <= 23.0)
		tmp = Float64(Float64(Float64(log(cosh(Float64(Float64(f * pi) * -0.25))) / pi) - Float64(log(sinh(t_0)) / pi)) * -4.0);
	else
		tmp = log((Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)) ^ Float64(-Float64(Float64(1.0 / pi) * 4.0))));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]}, Block[{t$95$1 = N[(N[(f * Pi), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, If[LessEqual[f, 23.0], N[(N[(N[(N[Log[N[Cosh[N[(N[(f * Pi), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] - N[(N[Log[N[Sinh[t$95$0], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[Log[N[Power[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision], (-N[(N[(1.0 / Pi), $MachinePrecision] * 4.0), $MachinePrecision])], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if f < 23

   1. Initial program 6.7%

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

   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 99.4%

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

      1. lift-/.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-cosh.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      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-sinh.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-log.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 99.4%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

   8. Applied rewrites 99.4%

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

   if 23 < f

   1. Initial program 3.0%

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

   2. Taylor expanded in f around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 1.7

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

   4. Applied rewrites 1.7%

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

   5. Taylor expanded in f around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 86.1

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in f around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 86.1

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

   10. Applied rewrites 86.1%

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

   11. Taylor expanded in f around 0

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

       1. *-commutative N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-*.f64 86.1

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

   13. Applied rewrites 86.1%

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

       1. lift-neg.f64 N/A

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

       2. lift-*.f64 N/A

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

   15. Applied rewrites 86.1%

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

Alternative 4: 99.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(f \cdot \pi\right) \cdot -0.25\\ t_1 := \mathsf{fma}\left(\pi \cdot f, 0.25, 1\right)\\ t_2 := e^{t\_0}\\ \mathbf{if}\;f \leq 23:\\ \;\;\;\;\left(\frac{\log \cosh t\_0}{\pi} - \frac{\log \sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}{\pi}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t\_1 + t\_2}{t\_1 - t\_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (* f PI) -0.25)) (t_1 (fma (* PI f) 0.25 1.0)) (t_2 (exp t_0)))
   (if (<= f 23.0)
     (* (- (/ (log (cosh t_0)) PI) (/ (log (sinh (* (* f PI) 0.25))) PI)) -4.0)
     (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2))))))))

C

double code(double f) {
	double t_0 = (f * ((double) M_PI)) * -0.25;
	double t_1 = fma((((double) M_PI) * f), 0.25, 1.0);
	double t_2 = exp(t_0);
	double tmp;
	if (f <= 23.0) {
		tmp = ((log(cosh(t_0)) / ((double) M_PI)) - (log(sinh(((f * ((double) M_PI)) * 0.25))) / ((double) M_PI))) * -4.0;
	} else {
		tmp = -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
	}
	return tmp;
}

Julia

function code(f)
	t_0 = Float64(Float64(f * pi) * -0.25)
	t_1 = fma(Float64(pi * f), 0.25, 1.0)
	t_2 = exp(t_0)
	tmp = 0.0
	if (f <= 23.0)
		tmp = Float64(Float64(Float64(log(cosh(t_0)) / pi) - Float64(log(sinh(Float64(Float64(f * pi) * 0.25))) / pi)) * -4.0);
	else
		tmp = Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(f * Pi), $MachinePrecision] * -0.25), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * f), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Exp[t$95$0], $MachinePrecision]}, If[LessEqual[f, 23.0], N[(N[(N[(N[Log[N[Cosh[t$95$0], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] - N[(N[Log[N[Sinh[N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] * -4.0), $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])]]]]

Derivation

1. Split input into 2 regimes

2. if f < 23

   1. Initial program 6.7%

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

   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 99.4%

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

      1. lift-/.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-cosh.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      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-sinh.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-log.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 99.4%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

   8. Applied rewrites 99.4%

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

   if 23 < f

   1. Initial program 3.0%

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

   2. Taylor expanded in f around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 1.7

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

   4. Applied rewrites 1.7%

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

   5. Taylor expanded in f around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 86.1

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in f around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 86.1

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

   10. Applied rewrites 86.1%

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

   11. Taylor expanded in f around 0

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

       1. *-commutative N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-*.f64 86.1

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

   13. Applied rewrites 86.1%

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

   14. Taylor expanded in f around 0

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

       1. *-commutative N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-*.f64 86.1

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

   16. Applied rewrites 86.1%

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

   17. Taylor expanded in f around 0

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

       1. *-commutative N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-*.f64 86.1

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

   19. Applied rewrites 86.1%

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

Alternative 5: 99.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(f \cdot \pi\right) \cdot 0.25\\ t_1 := e^{-t\_0}\\ \mathbf{if}\;f \leq 23:\\ \;\;\;\;\left(\frac{\log \cosh \left(\left(f \cdot \pi\right) \cdot -0.25\right)}{\pi} - \frac{\log \sinh t\_0}{\pi}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t\_0 + t\_1}{t\_0 - t\_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (* f PI) 0.25)) (t_1 (exp (- t_0))))
   (if (<= f 23.0)
     (*
      (- (/ (log (cosh (* (* f PI) -0.25))) PI) (/ (log (sinh t_0)) PI))
      -4.0)
     (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_0 t_1) (- t_0 t_1))))))))

C

double code(double f) {
	double t_0 = (f * ((double) M_PI)) * 0.25;
	double t_1 = exp(-t_0);
	double tmp;
	if (f <= 23.0) {
		tmp = ((log(cosh(((f * ((double) M_PI)) * -0.25))) / ((double) M_PI)) - (log(sinh(t_0)) / ((double) M_PI))) * -4.0;
	} else {
		tmp = -((1.0 / (((double) M_PI) / 4.0)) * log(((t_0 + t_1) / (t_0 - t_1))));
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = (f * Math.PI) * 0.25;
	double t_1 = Math.exp(-t_0);
	double tmp;
	if (f <= 23.0) {
		tmp = ((Math.log(Math.cosh(((f * Math.PI) * -0.25))) / Math.PI) - (Math.log(Math.sinh(t_0)) / Math.PI)) * -4.0;
	} else {
		tmp = -((1.0 / (Math.PI / 4.0)) * Math.log(((t_0 + t_1) / (t_0 - t_1))));
	}
	return tmp;
}

Python

def code(f):
	t_0 = (f * math.pi) * 0.25
	t_1 = math.exp(-t_0)
	tmp = 0
	if f <= 23.0:
		tmp = ((math.log(math.cosh(((f * math.pi) * -0.25))) / math.pi) - (math.log(math.sinh(t_0)) / math.pi)) * -4.0
	else:
		tmp = -((1.0 / (math.pi / 4.0)) * math.log(((t_0 + t_1) / (t_0 - t_1))))
	return tmp

Julia

function code(f)
	t_0 = Float64(Float64(f * pi) * 0.25)
	t_1 = exp(Float64(-t_0))
	tmp = 0.0
	if (f <= 23.0)
		tmp = Float64(Float64(Float64(log(cosh(Float64(Float64(f * pi) * -0.25))) / pi) - Float64(log(sinh(t_0)) / pi)) * -4.0);
	else
		tmp = Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_0 + t_1) / Float64(t_0 - t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	t_0 = (f * pi) * 0.25;
	t_1 = exp(-t_0);
	tmp = 0.0;
	if (f <= 23.0)
		tmp = ((log(cosh(((f * pi) * -0.25))) / pi) - (log(sinh(t_0)) / pi)) * -4.0;
	else
		tmp = -((1.0 / (pi / 4.0)) * log(((t_0 + t_1) / (t_0 - t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-t$95$0)], $MachinePrecision]}, If[LessEqual[f, 23.0], N[(N[(N[(N[Log[N[Cosh[N[(N[(f * Pi), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] - N[(N[Log[N[Sinh[t$95$0], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$0 + t$95$1), $MachinePrecision] / N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 2 regimes

2. if f < 23

   1. Initial program 6.7%

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

   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 99.4%

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

      1. lift-/.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-cosh.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      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-sinh.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-log.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 99.4%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

   8. Applied rewrites 99.4%

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

   if 23 < f

   1. Initial program 3.0%

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

   2. Taylor expanded in f around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 1.7

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

   4. Applied rewrites 1.7%

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

   5. Taylor expanded in f around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-PI.f64 86.1

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in f around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 86.1

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

   10. Applied rewrites 86.1%

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

   11. Taylor expanded in f around 0

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

       1. *-commutative N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-*.f64 86.1

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

   13. Applied rewrites 86.1%

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

   14. Taylor expanded in f around inf

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

       1. *-commutative N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-*.f64 3.0

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

   16. Applied rewrites 3.0%

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

   17. Taylor expanded in f around inf

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

       1. *-commutative N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-*.f64 86.1

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

   19. Applied rewrites 86.1%

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

Alternative 6: 99.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{4} \cdot \left(-f\right)}\\ \mathbf{if}\;f \leq 23:\\ \;\;\;\;\left(\frac{\log \cosh \left(\left(f \cdot \pi\right) \cdot -0.25\right)}{\pi} - \frac{\log \sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}{\pi}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(\frac{t\_0 + 1}{1 - t\_0}\right)}^{\left(-\frac{4}{\pi}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (* (/ PI 4.0) (- f)))))
   (if (<= f 23.0)
     (*
      (-
       (/ (log (cosh (* (* f PI) -0.25))) PI)
       (/ (log (sinh (* (* f PI) 0.25))) PI))
      -4.0)
     (log (pow (/ (+ t_0 1.0) (- 1.0 t_0)) (- (/ 4.0 PI)))))))

C

double code(double f) {
	double t_0 = exp(((((double) M_PI) / 4.0) * -f));
	double tmp;
	if (f <= 23.0) {
		tmp = ((log(cosh(((f * ((double) M_PI)) * -0.25))) / ((double) M_PI)) - (log(sinh(((f * ((double) M_PI)) * 0.25))) / ((double) M_PI))) * -4.0;
	} else {
		tmp = log(pow(((t_0 + 1.0) / (1.0 - t_0)), -(4.0 / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = Math.exp(((Math.PI / 4.0) * -f));
	double tmp;
	if (f <= 23.0) {
		tmp = ((Math.log(Math.cosh(((f * Math.PI) * -0.25))) / Math.PI) - (Math.log(Math.sinh(((f * Math.PI) * 0.25))) / Math.PI)) * -4.0;
	} else {
		tmp = Math.log(Math.pow(((t_0 + 1.0) / (1.0 - t_0)), -(4.0 / Math.PI)));
	}
	return tmp;
}

Python

def code(f):
	t_0 = math.exp(((math.pi / 4.0) * -f))
	tmp = 0
	if f <= 23.0:
		tmp = ((math.log(math.cosh(((f * math.pi) * -0.25))) / math.pi) - (math.log(math.sinh(((f * math.pi) * 0.25))) / math.pi)) * -4.0
	else:
		tmp = math.log(math.pow(((t_0 + 1.0) / (1.0 - t_0)), -(4.0 / math.pi)))
	return tmp

Julia

function code(f)
	t_0 = exp(Float64(Float64(pi / 4.0) * Float64(-f)))
	tmp = 0.0
	if (f <= 23.0)
		tmp = Float64(Float64(Float64(log(cosh(Float64(Float64(f * pi) * -0.25))) / pi) - Float64(log(sinh(Float64(Float64(f * pi) * 0.25))) / pi)) * -4.0);
	else
		tmp = log((Float64(Float64(t_0 + 1.0) / Float64(1.0 - t_0)) ^ Float64(-Float64(4.0 / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	t_0 = exp(((pi / 4.0) * -f));
	tmp = 0.0;
	if (f <= 23.0)
		tmp = ((log(cosh(((f * pi) * -0.25))) / pi) - (log(sinh(((f * pi) * 0.25))) / pi)) * -4.0;
	else
		tmp = log((((t_0 + 1.0) / (1.0 - t_0)) ^ -(4.0 / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := Block[{t$95$0 = N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * (-f)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[f, 23.0], N[(N[(N[(N[Log[N[Cosh[N[(N[(f * Pi), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] - N[(N[Log[N[Sinh[N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[Log[N[Power[N[(N[(t$95$0 + 1.0), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], (-N[(4.0 / Pi), $MachinePrecision])], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if f < 23

   1. Initial program 6.7%

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

   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 99.4%

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

      1. lift-/.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-cosh.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      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-sinh.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-log.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 99.4%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

   8. Applied rewrites 99.4%

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

   if 23 < f

   1. Initial program 3.0%

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

   2. Taylor expanded in f around 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 1.7%

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

   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 86.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-log.f64 N/A

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

   9. Applied rewrites 86.1%

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

Alternative 7: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(f \cdot \pi\right) \cdot -0.25\\ t_1 := e^{t\_0}\\ \mathbf{if}\;f \leq 23:\\ \;\;\;\;\left(\frac{\log \cosh t\_0}{\pi} - \frac{\log \sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}{\pi}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{\pi} \cdot \left(-\log \left(\frac{t\_1 + 1}{1 - t\_1}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (* f PI) -0.25)) (t_1 (exp t_0)))
   (if (<= f 23.0)
     (* (- (/ (log (cosh t_0)) PI) (/ (log (sinh (* (* f PI) 0.25))) PI)) -4.0)
     (* (/ 4.0 PI) (- (log (/ (+ t_1 1.0) (- 1.0 t_1))))))))

C

double code(double f) {
	double t_0 = (f * ((double) M_PI)) * -0.25;
	double t_1 = exp(t_0);
	double tmp;
	if (f <= 23.0) {
		tmp = ((log(cosh(t_0)) / ((double) M_PI)) - (log(sinh(((f * ((double) M_PI)) * 0.25))) / ((double) M_PI))) * -4.0;
	} else {
		tmp = (4.0 / ((double) M_PI)) * -log(((t_1 + 1.0) / (1.0 - t_1)));
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = (f * Math.PI) * -0.25;
	double t_1 = Math.exp(t_0);
	double tmp;
	if (f <= 23.0) {
		tmp = ((Math.log(Math.cosh(t_0)) / Math.PI) - (Math.log(Math.sinh(((f * Math.PI) * 0.25))) / Math.PI)) * -4.0;
	} else {
		tmp = (4.0 / Math.PI) * -Math.log(((t_1 + 1.0) / (1.0 - t_1)));
	}
	return tmp;
}

Python

def code(f):
	t_0 = (f * math.pi) * -0.25
	t_1 = math.exp(t_0)
	tmp = 0
	if f <= 23.0:
		tmp = ((math.log(math.cosh(t_0)) / math.pi) - (math.log(math.sinh(((f * math.pi) * 0.25))) / math.pi)) * -4.0
	else:
		tmp = (4.0 / math.pi) * -math.log(((t_1 + 1.0) / (1.0 - t_1)))
	return tmp

Julia

function code(f)
	t_0 = Float64(Float64(f * pi) * -0.25)
	t_1 = exp(t_0)
	tmp = 0.0
	if (f <= 23.0)
		tmp = Float64(Float64(Float64(log(cosh(t_0)) / pi) - Float64(log(sinh(Float64(Float64(f * pi) * 0.25))) / pi)) * -4.0);
	else
		tmp = Float64(Float64(4.0 / pi) * Float64(-log(Float64(Float64(t_1 + 1.0) / Float64(1.0 - t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	t_0 = (f * pi) * -0.25;
	t_1 = exp(t_0);
	tmp = 0.0;
	if (f <= 23.0)
		tmp = ((log(cosh(t_0)) / pi) - (log(sinh(((f * pi) * 0.25))) / pi)) * -4.0;
	else
		tmp = (4.0 / pi) * -log(((t_1 + 1.0) / (1.0 - t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(f * Pi), $MachinePrecision] * -0.25), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, If[LessEqual[f, 23.0], N[(N[(N[(N[Log[N[Cosh[t$95$0], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] - N[(N[Log[N[Sinh[N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(4.0 / Pi), $MachinePrecision] * (-N[Log[N[(N[(t$95$1 + 1.0), $MachinePrecision] / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if f < 23

   1. Initial program 6.7%

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

   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 99.4%

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

      1. lift-/.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-cosh.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      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-sinh.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-log.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 99.4%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

   8. Applied rewrites 99.4%

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

   if 23 < f

   1. Initial program 3.0%

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

   2. Taylor expanded in f around 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 1.7%

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

   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 86.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 86.1%

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

   10. Taylor expanded in f around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 86.1

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

   12. Applied rewrites 86.1%

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

   13. Taylor expanded in f around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 86.1

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

   15. Applied rewrites 86.1%

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

Alternative 8: 99.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(f \cdot \pi\right) \cdot -0.25}\\ t_1 := \left(f \cdot \pi\right) \cdot 0.25\\ \mathbf{if}\;f \leq 23:\\ \;\;\;\;\frac{\log \cosh t\_1 - \log \sinh t\_1}{\pi} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{\pi} \cdot \left(-\log \left(\frac{t\_0 + 1}{1 - t\_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (* (* f PI) -0.25))) (t_1 (* (* f PI) 0.25)))
   (if (<= f 23.0)
     (* (/ (- (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 <= 23.0) {
		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 <= 23.0) {
		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 <= 23.0:
		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 <= 23.0)
		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 <= 23.0)
		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, 23.0], 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 < 23

   1. Initial program 6.7%

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

   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 99.4%

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

      1. lift-/.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-cosh.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      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-sinh.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-log.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 99.4%

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

   if 23 < f

   1. Initial program 3.0%

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

   2. Taylor expanded in f around 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 1.7%

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

   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 86.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 86.1%

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

   10. Taylor expanded in f around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 86.1

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

   12. Applied rewrites 86.1%

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

   13. Taylor expanded in f around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 86.1

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

   15. Applied rewrites 86.1%

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

Alternative 9: 99.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(f \cdot \pi\right) \cdot -0.25}\\ t_1 := \left(f \cdot \pi\right) \cdot 0.25\\ \mathbf{if}\;f \leq 23:\\ \;\;\;\;\frac{\log \left(\frac{\cosh t\_1}{\sinh t\_1}\right) \cdot -4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{\pi} \cdot \left(-\log \left(\frac{t\_0 + 1}{1 - t\_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (* (* f PI) -0.25))) (t_1 (* (* f PI) 0.25)))
   (if (<= f 23.0)
     (/ (* (log (/ (cosh t_1) (sinh t_1))) -4.0) PI)
     (* (/ 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 <= 23.0) {
		tmp = (log((cosh(t_1) / sinh(t_1))) * -4.0) / ((double) M_PI);
	} 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 <= 23.0) {
		tmp = (Math.log((Math.cosh(t_1) / Math.sinh(t_1))) * -4.0) / Math.PI;
	} 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 <= 23.0:
		tmp = (math.log((math.cosh(t_1) / math.sinh(t_1))) * -4.0) / math.pi
	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 <= 23.0)
		tmp = Float64(Float64(log(Float64(cosh(t_1) / sinh(t_1))) * -4.0) / pi);
	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 <= 23.0)
		tmp = (log((cosh(t_1) / sinh(t_1))) * -4.0) / pi;
	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, 23.0], N[(N[(N[Log[N[(N[Cosh[t$95$1], $MachinePrecision] / N[Sinh[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -4.0), $MachinePrecision] / Pi), $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 < 23

   1. Initial program 6.7%

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

   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 99.4%

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

   6. Applied rewrites 99.4%

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

   if 23 < f

   1. Initial program 3.0%

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

   2. Taylor expanded in f around 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 1.7%

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

   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 86.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 86.1%

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

   10. Taylor expanded in f around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 86.1

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

   12. Applied rewrites 86.1%

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

   13. Taylor expanded in f around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 86.1

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

   15. Applied rewrites 86.1%

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

Alternative 10: 97.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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))) * -4.0) / Math.PI;
}

Python

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

Julia

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

MATLAB

function tmp = code(f)
	t_0 = (f * pi) * 0.25;
	tmp = (log((cosh(t_0) / sinh(t_0))) * -4.0) / pi;
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] * -4.0), $MachinePrecision] / Pi), $MachinePrecision]]

Derivation

1. Initial program 6.7%

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

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 97.2%

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

6. Applied rewrites 97.2%

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

Alternative 11: 96.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

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 97.2%

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

   1. lift-/.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-cosh.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   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-sinh.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-log.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 97.2%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

8. Applied rewrites 97.2%

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

9. Taylor expanded in f around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 N/A

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

    6. lift-PI.f64 96.6

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

11. Applied rewrites 96.6%

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

Alternative 12: 96.2% 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 6.7%

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

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 97.2%

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

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 96.2

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

7. Applied rewrites 96.2%

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

Alternative 13: 96.2% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

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 96.1%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)}{\pi} \cdot -4} \]
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. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. metadata-eval N/A

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

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

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

   11. log-div N/A

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

6. Applied rewrites 96.2%

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

Alternative 14: 96.1% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

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 96.1%

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

   1. lift-/.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. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. metadata-eval N/A

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

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

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-PI.f64 96.1

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

6. Applied rewrites 96.1%

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

Alternative 15: 96.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

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 97.2%

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

   1. lift-/.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-cosh.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   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-sinh.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-log.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 97.2%

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

7. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

9. Applied rewrites 96.1%

   \[\leadsto \frac{\log \left(\left(\left(0.5 \cdot \pi\right) \cdot 0.5\right) \cdot f\right)}{\pi} \cdot \color{blue}{4} \]
10. Add Preprocessing

Alternative 16: 96.1% 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 6.7%

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

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 96.1%

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

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 96.1

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

7. Applied rewrites 96.1%

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

Reproduce

herbie shell --seed 2025115 
(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)))))))))