VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 96.7%

Time: 11.6s

Alternatives: 7

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1579155288.3171306

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

3. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.8%

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

7. Applied rewrites 97.8%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-cosh.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sinh.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. log-div N/A

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

   12. lower--.f64 N/A

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

9. Applied rewrites 97.8%

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

    1. lift-log.f64 N/A

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

    2. lift-cosh.f64 N/A

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

    3. cosh-def N/A

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

    4. lift-*.f64 N/A

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

    5. lift-PI.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. *-commutative N/A

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

    8. lift-*.f64 N/A

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

    9. lift-PI.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. *-commutative N/A

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

    12. rec-exp N/A

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

11. Applied rewrites 97.9%

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

Alternative 2: 96.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

3. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.8%

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

7. Applied rewrites 97.8%

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

Alternative 3: 96.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

3. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.8%

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

7. Applied rewrites 97.8%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

9. Applied rewrites 97.7%

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

Alternative 4: 95.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, 0.125 \cdot \pi\right), f, 0\right), f, \frac{4}{\pi}\right)}{f}\right)}{\frac{-\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (log
   (/
    (fma
     (fma (fma (* PI 0.020833333333333332) -2.0 (* 0.125 PI)) f 0.0)
     f
     (/ 4.0 PI))
    f))
  (/ (- PI) 4.0)))

C

double code(double f) {
	return log((fma(fma(fma((((double) M_PI) * 0.020833333333333332), -2.0, (0.125 * ((double) M_PI))), f, 0.0), f, (4.0 / ((double) M_PI))) / f)) / (-((double) M_PI) / 4.0);
}

Julia

function code(f)
	return Float64(log(Float64(fma(fma(fma(Float64(pi * 0.020833333333333332), -2.0, Float64(0.125 * pi)), f, 0.0), f, Float64(4.0 / pi)) / f)) / Float64(Float64(-pi) / 4.0))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[(N[(N[(Pi * 0.020833333333333332), $MachinePrecision] * -2.0 + N[(0.125 * Pi), $MachinePrecision]), $MachinePrecision] * f + 0.0), $MachinePrecision] * f + N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / N[((-Pi) / 4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.1%

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

3. Taylor expanded in f around 0

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

5. Applied rewrites 95.6%

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

7. Applied rewrites 95.7%

   \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, 0.125 \cdot \pi\right), f, 0\right), f, \frac{4}{\pi}\right)}{f}\right)}{\frac{\pi}{4}}} \]

8. Final simplification 95.7%

   \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, 0.125 \cdot \pi\right), f, 0\right), f, \frac{4}{\pi}\right)}{f}\right)}{\frac{-\pi}{4}} \]
9. Add Preprocessing

Alternative 5: 95.8% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ -1.0 (/ PI 4.0))
  (log (/ (fma (/ 2.0 PI) 2.0 (* (* PI 0.08333333333333333) (* f f))) f))))

C

double code(double f) {
	return (-1.0 / (((double) M_PI) / 4.0)) * log((fma((2.0 / ((double) M_PI)), 2.0, ((((double) M_PI) * 0.08333333333333333) * (f * f))) / f));
}

Julia

function code(f)
	return Float64(Float64(-1.0 / Float64(pi / 4.0)) * log(Float64(fma(Float64(2.0 / pi), 2.0, Float64(Float64(pi * 0.08333333333333333) * Float64(f * f))) / f)))
end

Wolfram

code[f_] := N[(N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(N[(2.0 / Pi), $MachinePrecision] * 2.0 + N[(N[(Pi * 0.08333333333333333), $MachinePrecision] * N[(f * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.1%

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

3. Taylor expanded in f around 0

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

5. Applied rewrites 95.6%

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

6. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. distribute-rgt-out N/A

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

   4. lower-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. metadata-eval N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 95.6

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

8. Applied rewrites 95.6%

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

9. Final simplification 95.6%

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

Alternative 6: 95.5% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

3. Taylor expanded in f around 0

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

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

5. Applied rewrites 95.2%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. metadata-eval N/A

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

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

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

   11. log-div N/A

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

7. Applied rewrites 95.2%

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

Alternative 7: 95.5% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.1%

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

3. Taylor expanded in f around 0

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

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

5. Applied rewrites 95.2%

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

6. Taylor expanded in f around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-PI.f64 95.2

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

8. Applied rewrites 95.2%

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

Reproduce

herbie shell --seed 2025064 
(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)))))))))