VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 97.2%

Time: 8.4s

Alternatives: 7

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1554514533.4206696

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

2. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 97.2%

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

6. Applied rewrites 97.2%

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

Alternative 2: 96.3% accurate, 1.9× speedup

Math

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

FPCore

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

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)))) * -4.0) / ((double) M_PI);
}

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)))) * -4.0) / pi)
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] * -4.0), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 6.7%

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

2. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 97.2%

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

6. Applied rewrites 97.2%

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

7. 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. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-PI.f64 96.3

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

9. Applied rewrites 96.3%

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

Alternative 3: 96.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{\mathsf{fma}\left(0.03125 \cdot \left(f \cdot f\right), \pi \cdot \pi, 1\right)}{\left(\left(0.5 \cdot \pi\right) \cdot f\right) \cdot 0.5}\right) \cdot -4}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (*
   (log (/ (fma (* 0.03125 (* f f)) (* PI PI) 1.0) (* (* (* 0.5 PI) f) 0.5)))
   -4.0)
  PI))

C

double code(double f) {
	return (log((fma((0.03125 * (f * f)), (((double) M_PI) * ((double) M_PI)), 1.0) / (((0.5 * ((double) M_PI)) * f) * 0.5))) * -4.0) / ((double) M_PI);
}

Julia

function code(f)
	return Float64(Float64(log(Float64(fma(Float64(0.03125 * Float64(f * f)), Float64(pi * pi), 1.0) / Float64(Float64(Float64(0.5 * pi) * f) * 0.5))) * -4.0) / pi)
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[(N[(N[(0.5 * Pi), $MachinePrecision] * f), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -4.0), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 6.7%

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

2. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 97.2%

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

6. Applied rewrites 97.2%

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

7. 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. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-PI.f64 96.3

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

9. Applied rewrites 96.3%

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

10. Taylor expanded in f around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. *-commutative N/A

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

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

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

    5. metadata-eval N/A

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

    6. *-commutative N/A

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

    7. lift-*.f64 N/A

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

    8. lift-PI.f64 N/A

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

    9. lift-*.f64 96.0

       \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(0.03125 \cdot \left(f \cdot f\right), \pi \cdot \pi, 1\right)}{\left(\left(0.5 \cdot \pi\right) \cdot f\right) \cdot 0.5}\right) \cdot -4}{\pi} \]

12. Applied rewrites 96.0%

    \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(0.03125 \cdot \left(f \cdot f\right), \pi \cdot \pi, 1\right)}{\left(\left(0.5 \cdot \pi\right) \cdot f\right) \cdot 0.5}\right) \cdot -4}{\pi} \]
13. Add Preprocessing

Alternative 4: 95.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\log \left(\frac{2}{0.5 \cdot \pi}\right)}{\pi} + \frac{\log \left(\frac{1}{f}\right)}{\pi}\right) \cdot -4 \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (+ (/ (log (/ 2.0 (* 0.5 PI))) PI) (/ (log (/ 1.0 f)) PI)) -4.0))

C

double code(double f) {
	return ((log((2.0 / (0.5 * ((double) M_PI)))) / ((double) M_PI)) + (log((1.0 / f)) / ((double) M_PI))) * -4.0;
}

Java

public static double code(double f) {
	return ((Math.log((2.0 / (0.5 * Math.PI))) / Math.PI) + (Math.log((1.0 / f)) / Math.PI)) * -4.0;
}

Python

def code(f):
	return ((math.log((2.0 / (0.5 * math.pi))) / math.pi) + (math.log((1.0 / f)) / math.pi)) * -4.0

Julia

function code(f)
	return Float64(Float64(Float64(log(Float64(2.0 / Float64(0.5 * pi))) / pi) + Float64(log(Float64(1.0 / f)) / pi)) * -4.0)
end

MATLAB

function tmp = code(f)
	tmp = ((log((2.0 / (0.5 * pi))) / pi) + (log((1.0 / f)) / pi)) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[(N[Log[N[(2.0 / N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] + N[(N[Log[N[(1.0 / f), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 6.7%

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

2. Taylor expanded in f around 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 95.9%

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

   1. lift-PI.f64 N/A

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

6. Applied rewrites 95.9%

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

   1. lift-log.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. neg-log N/A

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

   4. lower-log.f64 N/A

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

   5. lower-/.f64 95.8

      \[\leadsto \left(\frac{\log \left(\frac{2}{0.5 \cdot \pi}\right)}{\pi} + \frac{\log \left(\frac{1}{f}\right)}{\pi}\right) \cdot -4 \]

8. Applied rewrites 95.8%

   \[\leadsto \left(\frac{\log \left(\frac{2}{0.5 \cdot \pi}\right)}{\pi} + \frac{\log \left(\frac{1}{f}\right)}{\pi}\right) \cdot -4 \]
9. Add Preprocessing

Alternative 5: 95.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

2. Taylor expanded in f around 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 95.9%

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

   1. lift-PI.f64 N/A

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

6. Applied rewrites 95.9%

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

   1. lift-+.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/r* N/A

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

   14. metadata-eval N/A

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

8. Applied rewrites 95.9%

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

Alternative 6: 95.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double f) {
	return ((Math.PI * Math.log(((4.0 / Math.PI) * (1.0 / f)))) * -4.0) / (Math.PI * Math.PI);
}

Python

def code(f):
	return ((math.pi * math.log(((4.0 / math.pi) * (1.0 / f)))) * -4.0) / (math.pi * math.pi)

Julia

function code(f)
	return Float64(Float64(Float64(pi * log(Float64(Float64(4.0 / pi) * Float64(1.0 / f)))) * -4.0) / Float64(pi * pi))
end

MATLAB

function tmp = code(f)
	tmp = ((pi * log(((4.0 / pi) * (1.0 / f)))) * -4.0) / (pi * pi);
end

Wolfram

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

Derivation

1. Initial program 6.7%

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

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 95.9%

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

   1. lift-PI.f64 N/A

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

6. Applied rewrites 95.9%

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

8. Applied rewrites 95.7%

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

   1. lift-*.f64 N/A

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

10. Applied rewrites 95.8%

    \[\leadsto \frac{\left(\pi \cdot \log \left(\frac{4}{\pi} \cdot \frac{1}{f}\right)\right) \cdot -4}{\color{blue}{\pi \cdot \pi}} \]
11. Add Preprocessing

Alternative 7: 95.8% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

2. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 95.9%

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

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 95.9

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

7. Applied rewrites 95.9%

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

Reproduce

herbie shell --seed 2025113 
(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)))))))))