VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 97.2%

Time: 9.0s

Alternatives: 5

Speedup: 4.6×

• could not determine a ground truth

  1. f = 1052592127.5552814

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: 6.8% 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: 97.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 97.2%

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

6. Applied rewrites 97.2%

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

Alternative 2: 96.5% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 97.2%

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

6. Applied rewrites 97.2%

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

7. Taylor expanded in f around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 96.5

      \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left({\left(f \cdot \pi\right)}^{2}, 0.03125, 1\right)}{\sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}\right) \cdot -4}{\pi} \]

9. Applied rewrites 96.5%

   \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left({\left(f \cdot \pi\right)}^{2}, 0.03125, 1\right)}{\sinh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}\right) \cdot -4}{\pi} \]
10. Step-by-step derivation

    1. lift-pow.f64 N/A

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

    2. lift-PI.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. unpow-prod-down N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 N/A

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

    7. pow2 N/A

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

    8. lift-*.f64 N/A

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

    9. lift-PI.f64 N/A

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

    10. lift-PI.f64 N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 96.5

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

11. Applied rewrites 96.5%

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

Alternative 3: 96.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 96.1%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. log-div N/A

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

   4. lower--.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. metadata-eval N/A

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

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

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

   11. *-commutative N/A

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

   12. lower-log.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

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

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

   16. metadata-eval N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lift-PI.f64 96.1

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

6. Applied rewrites 96.1%

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

   1. lift--.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. log-prod N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. unpow1 N/A

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

   9. metadata-eval N/A

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

   10. pow-pow N/A

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

   11. inv-pow N/A

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

   12. log-pow-rev N/A

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

   13. associate--r+ N/A

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

8. Applied rewrites 96.2%

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

Alternative 4: 96.1% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 96.1%

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

5. Taylor expanded in f around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-PI.f64 96.1

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

7. Applied rewrites 96.1%

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

Alternative 5: 1.6% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double f) {
	return (4.0 / Math.PI) * -Math.log(((0.08333333333333333 * Math.PI) * f));
}

Python

def code(f):
	return (4.0 / math.pi) * -math.log(((0.08333333333333333 * math.pi) * f))

Julia

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

MATLAB

function tmp = code(f)
	tmp = (4.0 / pi) * -log(((0.08333333333333333 * pi) * f));
end

Wolfram

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

Derivation

1. Initial program 6.8%

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

2. Taylor expanded in f around 0

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

4. Applied rewrites 96.4%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \mathsf{fma}\left(\frac{\pi}{\pi \cdot 0.5}, 0, \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi}, 0.125, -2 \cdot \frac{{\pi}^{3} \cdot 0.005208333333333333}{{\left(\pi \cdot 0.5\right)}^{2}}\right) \cdot f\right) \cdot f\right)}{f}\right)} \]

5. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. distribute-rgt-out N/A

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

   4. lower-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. metadata-eval 1.6

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

7. Applied rewrites 1.6%

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

   1. lift-neg.f64 N/A

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

9. Applied rewrites 1.6%

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

Reproduce

herbie shell --seed 2025092 
(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)))))))))