VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 97.0%

Time: 14.7s

Alternatives: 6

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1433149698.178225

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: 6.9% 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: 97.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

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 97.0%

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

6. Applied rewrites 97.0%

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

Alternative 2: 96.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

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 97.0%

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

6. Applied rewrites 97.0%

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

7. Taylor expanded in f around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-PI.f64 96.2

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

9. Applied rewrites 96.2%

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

Alternative 3: 96.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

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 97.0%

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

6. Applied rewrites 97.0%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-cosh.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sinh.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. log-div N/A

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

   12. lower--.f64 N/A

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

8. Applied rewrites 97.0%

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

9. Taylor expanded in f around 0

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

    1. associate-*r* N/A

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

    2. lower-*.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 N/A

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

    6. pow2 N/A

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

    7. lift-*.f64 N/A

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

    8. lift-PI.f64 N/A

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

    9. lift-PI.f64 96.2

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

11. Applied rewrites 96.2%

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

Alternative 4: 95.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{\mathsf{fma}\left(0.0625 \cdot \left(f \cdot f\right), \pi \cdot \pi, 2\right)}{\left(0.5 \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ (log (/ (fma (* 0.0625 (* f f)) (* PI PI) 2.0) (* (* 0.5 PI) f))) PI)
  -4.0))

C

double code(double f) {
	return (log((fma((0.0625 * (f * f)), (((double) M_PI) * ((double) M_PI)), 2.0) / ((0.5 * ((double) M_PI)) * f))) / ((double) M_PI)) * -4.0;
}

Julia

function code(f)
	return Float64(Float64(log(Float64(fma(Float64(0.0625 * Float64(f * f)), Float64(pi * pi), 2.0) / Float64(Float64(0.5 * pi) * f))) / pi) * -4.0)
end

Wolfram

code[f_] := N[(N[(N[Log[N[(N[(N[(0.0625 * N[(f * f), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * Pi), $MachinePrecision] * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

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 97.0%

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

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 95.8

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

7. Applied rewrites 95.8%

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

8. Taylor expanded in f around 0

   \[\leadsto \frac{\log \left(\frac{2 + \frac{1}{16} \cdot \left({f}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{\left(\frac{1}{2} \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]
9. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(\frac{1}{16} \cdot {f}^{2}, {\mathsf{PI}\left(\right)}^{2}, 2\right)}{\left(\frac{1}{2} \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(\frac{1}{16} \cdot {f}^{2}, {\mathsf{PI}\left(\right)}^{2}, 2\right)}{\left(\frac{1}{2} \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]

   5. unpow2 N/A

      \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(\frac{1}{16} \cdot \left(f \cdot f\right), {\mathsf{PI}\left(\right)}^{2}, 2\right)}{\left(\frac{1}{2} \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(\frac{1}{16} \cdot \left(f \cdot f\right), {\mathsf{PI}\left(\right)}^{2}, 2\right)}{\left(\frac{1}{2} \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]

   7. pow2 N/A

      \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(\frac{1}{16} \cdot \left(f \cdot f\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)}{\left(\frac{1}{2} \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(\frac{1}{16} \cdot \left(f \cdot f\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)}{\left(\frac{1}{2} \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]

   9. lift-PI.f64 N/A

      \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(\frac{1}{16} \cdot \left(f \cdot f\right), \pi \cdot \mathsf{PI}\left(\right), 2\right)}{\left(\frac{1}{2} \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]

   10. lift-PI.f64 95.8

       \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(0.0625 \cdot \left(f \cdot f\right), \pi \cdot \pi, 2\right)}{\left(0.5 \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]

10. Applied rewrites 95.8%

    \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(0.0625 \cdot \left(f \cdot f\right), \pi \cdot \pi, 2\right)}{\left(0.5 \cdot \pi\right) \cdot f}\right)}{\pi} \cdot -4 \]
11. Add Preprocessing

Alternative 5: 95.7% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

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 95.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. metadata-eval N/A

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

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

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-PI.f64 95.7

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

6. Applied rewrites 95.7%

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

Alternative 6: 95.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

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 95.7%

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

5. Taylor expanded in f around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-PI.f64 95.7

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

7. Applied rewrites 95.7%

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

Reproduce

herbie shell --seed 2025124 
(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)))))))))