VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.0%

Time: 10.6s

Alternatives: 12

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1405983773.1682036

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot f, 0.0026041666666666665, \left(\pi \cdot \pi\right) \cdot 0.03125\right), f, \pi \cdot 0.25\right), f, 1\right)\\ t_1 := \left(\pi \cdot f\right) \cdot -0.25\\ t_2 := e^{t\_1}\\ \mathbf{if}\;f \leq 23.5:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{\cosh t\_1}{\sinh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{\pi} \cdot \left(-\log \left(\frac{t\_0 + t\_2}{t\_0 - t\_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0
         (fma
          (fma
           (fma
            (* (* (* PI PI) PI) f)
            0.0026041666666666665
            (* (* PI PI) 0.03125))
           f
           (* PI 0.25))
          f
          1.0))
        (t_1 (* (* PI f) -0.25))
        (t_2 (exp t_1)))
   (if (<= f 23.5)
     (* -4.0 (/ (log (/ (cosh t_1) (sinh (* (* 0.25 f) PI)))) PI))
     (* (/ 4.0 PI) (- (log (/ (+ t_0 t_2) (- t_0 t_2))))))))

C

double code(double f) {
	double t_0 = fma(fma(fma((((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * f), 0.0026041666666666665, ((((double) M_PI) * ((double) M_PI)) * 0.03125)), f, (((double) M_PI) * 0.25)), f, 1.0);
	double t_1 = (((double) M_PI) * f) * -0.25;
	double t_2 = exp(t_1);
	double tmp;
	if (f <= 23.5) {
		tmp = -4.0 * (log((cosh(t_1) / sinh(((0.25 * f) * ((double) M_PI))))) / ((double) M_PI));
	} else {
		tmp = (4.0 / ((double) M_PI)) * -log(((t_0 + t_2) / (t_0 - t_2)));
	}
	return tmp;
}

Julia

function code(f)
	t_0 = fma(fma(fma(Float64(Float64(Float64(pi * pi) * pi) * f), 0.0026041666666666665, Float64(Float64(pi * pi) * 0.03125)), f, Float64(pi * 0.25)), f, 1.0)
	t_1 = Float64(Float64(pi * f) * -0.25)
	t_2 = exp(t_1)
	tmp = 0.0
	if (f <= 23.5)
		tmp = Float64(-4.0 * Float64(log(Float64(cosh(t_1) / sinh(Float64(Float64(0.25 * f) * pi)))) / pi));
	else
		tmp = Float64(Float64(4.0 / pi) * Float64(-log(Float64(Float64(t_0 + t_2) / Float64(t_0 - t_2)))));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * f), $MachinePrecision] * 0.0026041666666666665 + N[(N[(Pi * Pi), $MachinePrecision] * 0.03125), $MachinePrecision]), $MachinePrecision] * f + N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision] * f + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * f), $MachinePrecision] * -0.25), $MachinePrecision]}, Block[{t$95$2 = N[Exp[t$95$1], $MachinePrecision]}, If[LessEqual[f, 23.5], N[(-4.0 * N[(N[Log[N[(N[Cosh[t$95$1], $MachinePrecision] / N[Sinh[N[(N[(0.25 * f), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(4.0 / Pi), $MachinePrecision] * (-N[Log[N[(N[(t$95$0 + t$95$2), $MachinePrecision] / N[(t$95$0 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if f < 23.5

   1. Initial program 6.7%

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

   3. Applied rewrites 96.8%

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

   5. Applied rewrites 96.8%

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

   if 23.5 < f

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 6.0

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

   4. Applied rewrites 6.0%

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

   5. Taylor expanded in f around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 8.0

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

   7. Applied rewrites 8.0%

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

   9. Applied rewrites 8.0%

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

   10. Taylor expanded in f around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

   12. Applied rewrites 5.9%

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

   13. Taylor expanded in f around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

   15. Applied rewrites 8.2%

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

Alternative 2: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, 0.03125, \pi \cdot 0.25\right), f, 1\right)\\ t_1 := \left(\pi \cdot f\right) \cdot -0.25\\ t_2 := e^{t\_1}\\ \mathbf{if}\;f \leq 23.5:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{\cosh t\_1}{\sinh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(\frac{t\_0 + t\_2}{t\_0 - t\_2}\right)}^{\left(-\frac{4}{\pi}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (fma (fma (* (* PI PI) f) 0.03125 (* PI 0.25)) f 1.0))
        (t_1 (* (* PI f) -0.25))
        (t_2 (exp t_1)))
   (if (<= f 23.5)
     (* -4.0 (/ (log (/ (cosh t_1) (sinh (* (* 0.25 f) PI)))) PI))
     (log (pow (/ (+ t_0 t_2) (- t_0 t_2)) (- (/ 4.0 PI)))))))

C

double code(double f) {
	double t_0 = fma(fma(((((double) M_PI) * ((double) M_PI)) * f), 0.03125, (((double) M_PI) * 0.25)), f, 1.0);
	double t_1 = (((double) M_PI) * f) * -0.25;
	double t_2 = exp(t_1);
	double tmp;
	if (f <= 23.5) {
		tmp = -4.0 * (log((cosh(t_1) / sinh(((0.25 * f) * ((double) M_PI))))) / ((double) M_PI));
	} else {
		tmp = log(pow(((t_0 + t_2) / (t_0 - t_2)), -(4.0 / ((double) M_PI))));
	}
	return tmp;
}

Julia

function code(f)
	t_0 = fma(fma(Float64(Float64(pi * pi) * f), 0.03125, Float64(pi * 0.25)), f, 1.0)
	t_1 = Float64(Float64(pi * f) * -0.25)
	t_2 = exp(t_1)
	tmp = 0.0
	if (f <= 23.5)
		tmp = Float64(-4.0 * Float64(log(Float64(cosh(t_1) / sinh(Float64(Float64(0.25 * f) * pi)))) / pi));
	else
		tmp = log((Float64(Float64(t_0 + t_2) / Float64(t_0 - t_2)) ^ Float64(-Float64(4.0 / pi))));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * f), $MachinePrecision] * 0.03125 + N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision] * f + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * f), $MachinePrecision] * -0.25), $MachinePrecision]}, Block[{t$95$2 = N[Exp[t$95$1], $MachinePrecision]}, If[LessEqual[f, 23.5], N[(-4.0 * N[(N[Log[N[(N[Cosh[t$95$1], $MachinePrecision] / N[Sinh[N[(N[(0.25 * f), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[Log[N[Power[N[(N[(t$95$0 + t$95$2), $MachinePrecision] / N[(t$95$0 - t$95$2), $MachinePrecision]), $MachinePrecision], (-N[(4.0 / Pi), $MachinePrecision])], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if f < 23.5

   1. Initial program 6.7%

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

   3. Applied rewrites 96.8%

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

   5. Applied rewrites 96.8%

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

   if 23.5 < f

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 6.0

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

   4. Applied rewrites 6.0%

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

   5. Taylor expanded in f around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 8.0

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

   7. Applied rewrites 8.0%

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

   9. Applied rewrites 8.0%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot f, 0.03125, \pi \cdot 0.25\right), f, 1\right)\\ t_1 := \left(\pi \cdot f\right) \cdot -0.25\\ t_2 := e^{t\_1}\\ \mathbf{if}\;f \leq 23.5:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{\cosh t\_1}{\sinh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{\pi} \cdot \left(-\log \left(\frac{t\_0 + t\_2}{t\_0 - t\_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (fma (fma (* (* PI PI) f) 0.03125 (* PI 0.25)) f 1.0))
        (t_1 (* (* PI f) -0.25))
        (t_2 (exp t_1)))
   (if (<= f 23.5)
     (* -4.0 (/ (log (/ (cosh t_1) (sinh (* (* 0.25 f) PI)))) PI))
     (* (/ 4.0 PI) (- (log (/ (+ t_0 t_2) (- t_0 t_2))))))))

C

double code(double f) {
	double t_0 = fma(fma(((((double) M_PI) * ((double) M_PI)) * f), 0.03125, (((double) M_PI) * 0.25)), f, 1.0);
	double t_1 = (((double) M_PI) * f) * -0.25;
	double t_2 = exp(t_1);
	double tmp;
	if (f <= 23.5) {
		tmp = -4.0 * (log((cosh(t_1) / sinh(((0.25 * f) * ((double) M_PI))))) / ((double) M_PI));
	} else {
		tmp = (4.0 / ((double) M_PI)) * -log(((t_0 + t_2) / (t_0 - t_2)));
	}
	return tmp;
}

Julia

function code(f)
	t_0 = fma(fma(Float64(Float64(pi * pi) * f), 0.03125, Float64(pi * 0.25)), f, 1.0)
	t_1 = Float64(Float64(pi * f) * -0.25)
	t_2 = exp(t_1)
	tmp = 0.0
	if (f <= 23.5)
		tmp = Float64(-4.0 * Float64(log(Float64(cosh(t_1) / sinh(Float64(Float64(0.25 * f) * pi)))) / pi));
	else
		tmp = Float64(Float64(4.0 / pi) * Float64(-log(Float64(Float64(t_0 + t_2) / Float64(t_0 - t_2)))));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * f), $MachinePrecision] * 0.03125 + N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision] * f + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * f), $MachinePrecision] * -0.25), $MachinePrecision]}, Block[{t$95$2 = N[Exp[t$95$1], $MachinePrecision]}, If[LessEqual[f, 23.5], N[(-4.0 * N[(N[Log[N[(N[Cosh[t$95$1], $MachinePrecision] / N[Sinh[N[(N[(0.25 * f), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(4.0 / Pi), $MachinePrecision] * (-N[Log[N[(N[(t$95$0 + t$95$2), $MachinePrecision] / N[(t$95$0 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if f < 23.5

   1. Initial program 6.7%

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

   3. Applied rewrites 96.8%

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

   5. Applied rewrites 96.8%

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

   if 23.5 < f

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 6.0

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

   4. Applied rewrites 6.0%

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

   5. Taylor expanded in f around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 8.0

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

   7. Applied rewrites 8.0%

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

   9. Applied rewrites 8.0%

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-0.25 \cdot \left(\pi \cdot f\right)}\\ t_1 := \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot f\right) \cdot 0.03125, f, 1\right)\\ \mathbf{if}\;f \leq 23.5:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{\cosh \left(\left(\pi \cdot f\right) \cdot -0.25\right)}{\sinh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;-\frac{1}{\frac{\pi}{4}} \cdot \left(-\log \left(\frac{t\_1 - t\_0}{t\_0 + t\_1}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (* -0.25 (* PI f))))
        (t_1 (fma (* (* (* PI PI) f) 0.03125) f 1.0)))
   (if (<= f 23.5)
     (*
      -4.0
      (/ (log (/ (cosh (* (* PI f) -0.25)) (sinh (* (* 0.25 f) PI)))) PI))
     (- (* (/ 1.0 (/ PI 4.0)) (- (log (/ (- t_1 t_0) (+ t_0 t_1)))))))))

C

double code(double f) {
	double t_0 = exp((-0.25 * (((double) M_PI) * f)));
	double t_1 = fma((((((double) M_PI) * ((double) M_PI)) * f) * 0.03125), f, 1.0);
	double tmp;
	if (f <= 23.5) {
		tmp = -4.0 * (log((cosh(((((double) M_PI) * f) * -0.25)) / sinh(((0.25 * f) * ((double) M_PI))))) / ((double) M_PI));
	} else {
		tmp = -((1.0 / (((double) M_PI) / 4.0)) * -log(((t_1 - t_0) / (t_0 + t_1))));
	}
	return tmp;
}

Julia

function code(f)
	t_0 = exp(Float64(-0.25 * Float64(pi * f)))
	t_1 = fma(Float64(Float64(Float64(pi * pi) * f) * 0.03125), f, 1.0)
	tmp = 0.0
	if (f <= 23.5)
		tmp = Float64(-4.0 * Float64(log(Float64(cosh(Float64(Float64(pi * f) * -0.25)) / sinh(Float64(Float64(0.25 * f) * pi)))) / pi));
	else
		tmp = Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * Float64(-log(Float64(Float64(t_1 - t_0) / Float64(t_0 + t_1))))));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[Exp[N[(-0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * f), $MachinePrecision] * 0.03125), $MachinePrecision] * f + 1.0), $MachinePrecision]}, If[LessEqual[f, 23.5], N[(-4.0 * N[(N[Log[N[(N[Cosh[N[(N[(Pi * f), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] / N[Sinh[N[(N[(0.25 * f), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * (-N[Log[N[(N[(t$95$1 - t$95$0), $MachinePrecision] / N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision])]]]

Derivation

1. Split input into 2 regimes

2. if f < 23.5

   1. Initial program 6.7%

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

   3. Applied rewrites 96.8%

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

   5. Applied rewrites 96.8%

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

   if 23.5 < f

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 6.0

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

   4. Applied rewrites 6.0%

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

   5. Taylor expanded in f around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 8.0

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

   7. Applied rewrites 8.0%

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

   8. Taylor expanded in f around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 5.4

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

   10. Applied rewrites 5.4%

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

   11. Taylor expanded in f around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-PI.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. lift-*.f64 6.4

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

   13. Applied rewrites 6.4%

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

   15. Applied rewrites 6.4%

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

Alternative 5: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-0.25 \cdot \left(\pi \cdot f\right)}\\ t_1 := \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot f\right) \cdot 0.03125, f, 1\right)\\ \mathbf{if}\;f \leq 23.5:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{\cosh \left(\left(\pi \cdot f\right) \cdot -0.25\right)}{\sinh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{\pi} \cdot \left(-\log \left(\frac{t\_0 + t\_1}{t\_1 - t\_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (* -0.25 (* PI f))))
        (t_1 (fma (* (* (* PI PI) f) 0.03125) f 1.0)))
   (if (<= f 23.5)
     (*
      -4.0
      (/ (log (/ (cosh (* (* PI f) -0.25)) (sinh (* (* 0.25 f) PI)))) PI))
     (* (/ 4.0 PI) (- (log (/ (+ t_0 t_1) (- t_1 t_0))))))))

C

double code(double f) {
	double t_0 = exp((-0.25 * (((double) M_PI) * f)));
	double t_1 = fma((((((double) M_PI) * ((double) M_PI)) * f) * 0.03125), f, 1.0);
	double tmp;
	if (f <= 23.5) {
		tmp = -4.0 * (log((cosh(((((double) M_PI) * f) * -0.25)) / sinh(((0.25 * f) * ((double) M_PI))))) / ((double) M_PI));
	} 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(-0.25 * Float64(pi * f)))
	t_1 = fma(Float64(Float64(Float64(pi * pi) * f) * 0.03125), f, 1.0)
	tmp = 0.0
	if (f <= 23.5)
		tmp = Float64(-4.0 * Float64(log(Float64(cosh(Float64(Float64(pi * f) * -0.25)) / sinh(Float64(Float64(0.25 * f) * pi)))) / pi));
	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[(-0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * f), $MachinePrecision] * 0.03125), $MachinePrecision] * f + 1.0), $MachinePrecision]}, If[LessEqual[f, 23.5], N[(-4.0 * N[(N[Log[N[(N[Cosh[N[(N[(Pi * f), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] / N[Sinh[N[(N[(0.25 * f), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $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 < 23.5

   1. Initial program 6.7%

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

   3. Applied rewrites 96.8%

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

   5. Applied rewrites 96.8%

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

   if 23.5 < f

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 6.0

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

   4. Applied rewrites 6.0%

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

   5. Taylor expanded in f around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 8.0

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

   7. Applied rewrites 8.0%

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

   8. Taylor expanded in f around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 5.4

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

   10. Applied rewrites 5.4%

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

   11. Taylor expanded in f around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-PI.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. lift-*.f64 6.4

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

   13. Applied rewrites 6.4%

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

   15. Applied rewrites 6.4%

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

Alternative 6: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.03125 \cdot \left(f \cdot f\right)\right) \cdot \left(\pi \cdot \pi\right)\\ t_1 := \left(\pi \cdot f\right) \cdot -0.25\\ t_2 := e^{t\_1}\\ \mathbf{if}\;f \leq 23.5:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{\cosh t\_1}{\sinh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{\pi} \cdot \left(-\log \left(\frac{t\_0 + t\_2}{t\_0 - t\_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (* 0.03125 (* f f)) (* PI PI)))
        (t_1 (* (* PI f) -0.25))
        (t_2 (exp t_1)))
   (if (<= f 23.5)
     (* -4.0 (/ (log (/ (cosh t_1) (sinh (* (* 0.25 f) PI)))) PI))
     (* (/ 4.0 PI) (- (log (/ (+ t_0 t_2) (- t_0 t_2))))))))

C

double code(double f) {
	double t_0 = (0.03125 * (f * f)) * (((double) M_PI) * ((double) M_PI));
	double t_1 = (((double) M_PI) * f) * -0.25;
	double t_2 = exp(t_1);
	double tmp;
	if (f <= 23.5) {
		tmp = -4.0 * (log((cosh(t_1) / sinh(((0.25 * f) * ((double) M_PI))))) / ((double) M_PI));
	} else {
		tmp = (4.0 / ((double) M_PI)) * -log(((t_0 + t_2) / (t_0 - t_2)));
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = (0.03125 * (f * f)) * (Math.PI * Math.PI);
	double t_1 = (Math.PI * f) * -0.25;
	double t_2 = Math.exp(t_1);
	double tmp;
	if (f <= 23.5) {
		tmp = -4.0 * (Math.log((Math.cosh(t_1) / Math.sinh(((0.25 * f) * Math.PI)))) / Math.PI);
	} else {
		tmp = (4.0 / Math.PI) * -Math.log(((t_0 + t_2) / (t_0 - t_2)));
	}
	return tmp;
}

Python

def code(f):
	t_0 = (0.03125 * (f * f)) * (math.pi * math.pi)
	t_1 = (math.pi * f) * -0.25
	t_2 = math.exp(t_1)
	tmp = 0
	if f <= 23.5:
		tmp = -4.0 * (math.log((math.cosh(t_1) / math.sinh(((0.25 * f) * math.pi)))) / math.pi)
	else:
		tmp = (4.0 / math.pi) * -math.log(((t_0 + t_2) / (t_0 - t_2)))
	return tmp

Julia

function code(f)
	t_0 = Float64(Float64(0.03125 * Float64(f * f)) * Float64(pi * pi))
	t_1 = Float64(Float64(pi * f) * -0.25)
	t_2 = exp(t_1)
	tmp = 0.0
	if (f <= 23.5)
		tmp = Float64(-4.0 * Float64(log(Float64(cosh(t_1) / sinh(Float64(Float64(0.25 * f) * pi)))) / pi));
	else
		tmp = Float64(Float64(4.0 / pi) * Float64(-log(Float64(Float64(t_0 + t_2) / Float64(t_0 - t_2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	t_0 = (0.03125 * (f * f)) * (pi * pi);
	t_1 = (pi * f) * -0.25;
	t_2 = exp(t_1);
	tmp = 0.0;
	if (f <= 23.5)
		tmp = -4.0 * (log((cosh(t_1) / sinh(((0.25 * f) * pi)))) / pi);
	else
		tmp = (4.0 / pi) * -log(((t_0 + t_2) / (t_0 - t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(0.03125 * N[(f * f), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * f), $MachinePrecision] * -0.25), $MachinePrecision]}, Block[{t$95$2 = N[Exp[t$95$1], $MachinePrecision]}, If[LessEqual[f, 23.5], N[(-4.0 * N[(N[Log[N[(N[Cosh[t$95$1], $MachinePrecision] / N[Sinh[N[(N[(0.25 * f), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(4.0 / Pi), $MachinePrecision] * (-N[Log[N[(N[(t$95$0 + t$95$2), $MachinePrecision] / N[(t$95$0 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if f < 23.5

   1. Initial program 6.7%

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

   3. Applied rewrites 96.8%

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

   5. Applied rewrites 96.8%

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

   if 23.5 < f

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 6.0

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

   4. Applied rewrites 6.0%

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

   5. Taylor expanded in f around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 8.0

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

   7. Applied rewrites 8.0%

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

   9. Applied rewrites 8.0%

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

   10. Taylor expanded in f around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-PI.f64 N/A

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

       9. lift-PI.f64 4.2

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

   12. Applied rewrites 4.2%

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

   13. Taylor expanded in f around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-PI.f64 N/A

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

       9. lift-PI.f64 2.3

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

   15. Applied rewrites 2.3%

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

Alternative 7: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(f \cdot 0.25, \pi, 1\right)\\ t_1 := \left(\pi \cdot f\right) \cdot -0.25\\ t_2 := e^{t\_1}\\ \mathbf{if}\;f \leq 23.5:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{\cosh t\_1}{\sinh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{\pi} \cdot \left(-\log \left(\frac{t\_0 + t\_2}{t\_0 - t\_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (fma (* f 0.25) PI 1.0)) (t_1 (* (* PI f) -0.25)) (t_2 (exp t_1)))
   (if (<= f 23.5)
     (* -4.0 (/ (log (/ (cosh t_1) (sinh (* (* 0.25 f) PI)))) PI))
     (* (/ 4.0 PI) (- (log (/ (+ t_0 t_2) (- t_0 t_2))))))))

C

double code(double f) {
	double t_0 = fma((f * 0.25), ((double) M_PI), 1.0);
	double t_1 = (((double) M_PI) * f) * -0.25;
	double t_2 = exp(t_1);
	double tmp;
	if (f <= 23.5) {
		tmp = -4.0 * (log((cosh(t_1) / sinh(((0.25 * f) * ((double) M_PI))))) / ((double) M_PI));
	} else {
		tmp = (4.0 / ((double) M_PI)) * -log(((t_0 + t_2) / (t_0 - t_2)));
	}
	return tmp;
}

Julia

function code(f)
	t_0 = fma(Float64(f * 0.25), pi, 1.0)
	t_1 = Float64(Float64(pi * f) * -0.25)
	t_2 = exp(t_1)
	tmp = 0.0
	if (f <= 23.5)
		tmp = Float64(-4.0 * Float64(log(Float64(cosh(t_1) / sinh(Float64(Float64(0.25 * f) * pi)))) / pi));
	else
		tmp = Float64(Float64(4.0 / pi) * Float64(-log(Float64(Float64(t_0 + t_2) / Float64(t_0 - t_2)))));
	end
	return tmp
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(f * 0.25), $MachinePrecision] * Pi + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * f), $MachinePrecision] * -0.25), $MachinePrecision]}, Block[{t$95$2 = N[Exp[t$95$1], $MachinePrecision]}, If[LessEqual[f, 23.5], N[(-4.0 * N[(N[Log[N[(N[Cosh[t$95$1], $MachinePrecision] / N[Sinh[N[(N[(0.25 * f), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(4.0 / Pi), $MachinePrecision] * (-N[Log[N[(N[(t$95$0 + t$95$2), $MachinePrecision] / N[(t$95$0 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if f < 23.5

   1. Initial program 6.7%

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

   3. Applied rewrites 96.8%

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

   5. Applied rewrites 96.8%

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

   if 23.5 < f

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 6.0

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

   4. Applied rewrites 6.0%

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

   5. Taylor expanded in f around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 8.0

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

   7. Applied rewrites 8.0%

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

   9. Applied rewrites 8.0%

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

   10. Taylor expanded in f around 0

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

       1. +-commutative N/A

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

       2. associate-*l* N/A

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

       3. lower-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lift-PI.f64 5.8

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

   12. Applied rewrites 5.8%

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

   13. Taylor expanded in f around 0

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

       1. +-commutative N/A

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

       2. associate-*l* N/A

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

       3. lower-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lift-PI.f64 7.6

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

   15. Applied rewrites 7.6%

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

Alternative 8: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\pi}{4} \cdot f}\\ \mathbf{if}\;f \leq 23.5:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{\cosh \left(\left(\pi \cdot f\right) \cdot -0.25\right)}{\sinh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{1 + t\_0}{1 - t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (- (* (/ PI 4.0) f)))))
   (if (<= f 23.5)
     (*
      -4.0
      (/ (log (/ (cosh (* (* PI f) -0.25)) (sinh (* (* 0.25 f) PI)))) PI))
     (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ 1.0 t_0) (- 1.0 t_0))))))))

C

double code(double f) {
	double t_0 = exp(-((((double) M_PI) / 4.0) * f));
	double tmp;
	if (f <= 23.5) {
		tmp = -4.0 * (log((cosh(((((double) M_PI) * f) * -0.25)) / sinh(((0.25 * f) * ((double) M_PI))))) / ((double) M_PI));
	} else {
		tmp = -((1.0 / (((double) M_PI) / 4.0)) * log(((1.0 + t_0) / (1.0 - t_0))));
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = Math.exp(-((Math.PI / 4.0) * f));
	double tmp;
	if (f <= 23.5) {
		tmp = -4.0 * (Math.log((Math.cosh(((Math.PI * f) * -0.25)) / Math.sinh(((0.25 * f) * Math.PI)))) / Math.PI);
	} else {
		tmp = -((1.0 / (Math.PI / 4.0)) * Math.log(((1.0 + t_0) / (1.0 - t_0))));
	}
	return tmp;
}

Python

def code(f):
	t_0 = math.exp(-((math.pi / 4.0) * f))
	tmp = 0
	if f <= 23.5:
		tmp = -4.0 * (math.log((math.cosh(((math.pi * f) * -0.25)) / math.sinh(((0.25 * f) * math.pi)))) / math.pi)
	else:
		tmp = -((1.0 / (math.pi / 4.0)) * math.log(((1.0 + t_0) / (1.0 - t_0))))
	return tmp

Julia

function code(f)
	t_0 = exp(Float64(-Float64(Float64(pi / 4.0) * f)))
	tmp = 0.0
	if (f <= 23.5)
		tmp = Float64(-4.0 * Float64(log(Float64(cosh(Float64(Float64(pi * f) * -0.25)) / sinh(Float64(Float64(0.25 * f) * pi)))) / pi));
	else
		tmp = Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(1.0 + t_0) / Float64(1.0 - t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	t_0 = exp(-((pi / 4.0) * f));
	tmp = 0.0;
	if (f <= 23.5)
		tmp = -4.0 * (log((cosh(((pi * f) * -0.25)) / sinh(((0.25 * f) * pi)))) / pi);
	else
		tmp = -((1.0 / (pi / 4.0)) * log(((1.0 + t_0) / (1.0 - t_0))));
	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, 23.5], N[(-4.0 * N[(N[Log[N[(N[Cosh[N[(N[(Pi * f), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] / N[Sinh[N[(N[(0.25 * f), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(1.0 + t$95$0), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if f < 23.5

   1. Initial program 6.7%

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

   3. Applied rewrites 96.8%

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

   5. Applied rewrites 96.8%

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

   if 23.5 < f

   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

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

   4. Applied rewrites 5.3%

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

   5. Taylor expanded in f around 0

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

   7. Applied rewrites 6.4%

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

Alternative 9: 96.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

3. Applied rewrites 96.8%

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

5. Applied rewrites 96.8%

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

Alternative 10: 96.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{\mathsf{fma}\left(0.03125 \cdot \left(f \cdot f\right), \pi \cdot \pi, 1\right)}{\sinh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log (/ (fma (* 0.03125 (* f f)) (* PI PI) 1.0) (sinh (* (* 0.25 f) PI))))
   PI)))

C

double code(double f) {
	return -4.0 * (log((fma((0.03125 * (f * f)), (((double) M_PI) * ((double) M_PI)), 1.0) / sinh(((0.25 * f) * ((double) M_PI))))) / ((double) M_PI));
}

Julia

function code(f)
	return Float64(-4.0 * Float64(log(Float64(fma(Float64(0.03125 * Float64(f * f)), Float64(pi * pi), 1.0) / sinh(Float64(Float64(0.25 * f) * pi)))) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(N[(N[(0.03125 * N[(f * f), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sinh[N[(N[(0.25 * f), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.7%

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

3. Applied rewrites 96.8%

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

5. Applied rewrites 96.8%

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

6. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. cosh-neg-rev N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. +-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-PI.f64 N/A

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

   17. lift-PI.f64 96.2

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

8. Applied rewrites 96.2%

   \[\leadsto -4 \cdot \frac{\log \left(\frac{\color{blue}{\mathsf{fma}\left(0.03125 \cdot \left(f \cdot f\right), \pi \cdot \pi, 1\right)}}{\sinh \left(\left(0.25 \cdot f\right) \cdot \pi\right)}\right)}{\pi} \]
9. Add Preprocessing

Alternative 11: 95.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

3. Applied rewrites 96.8%

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

4. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

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

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 95.8

      \[\leadsto -4 \cdot \frac{\log \left(1 \cdot \frac{\cosh \left(\left(0.25 \cdot \pi\right) \cdot f\right)}{\left(\left(0.5 \cdot \pi\right) \cdot f\right) \cdot 0.5}\right)}{\pi} \]

6. Applied rewrites 95.8%

   \[\leadsto -4 \cdot \frac{\log \left(1 \cdot \frac{\cosh \left(\left(0.25 \cdot \pi\right) \cdot f\right)}{\color{blue}{\left(\left(0.5 \cdot \pi\right) \cdot f\right) \cdot 0.5}}\right)}{\pi} \]
7. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. lower-*.f64 95.8

      \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(1 \cdot \frac{\cosh \left(\left(0.25 \cdot \pi\right) \cdot f\right)}{\left(\left(0.5 \cdot \pi\right) \cdot f\right) \cdot 0.5}\right)}{\pi}} \]

8. Applied rewrites 95.8%

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

Alternative 12: 95.8% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(N[(N[Log[N[(4.0 / N[(Pi * 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 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 95.8%

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

5. Taylor expanded in f around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lift-PI.f64 95.8

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

7. Applied rewrites 95.8%

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

Reproduce

herbie shell --seed 2025132 
(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)))))))))