VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 97.3%

Time: 10.0s

Alternatives: 4

Speedup: 4.6×

• could not determine a ground truth

  1. f = 965561970.7771459

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: 97.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.3%

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

Alternative 2: 96.5% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

3. Taylor expanded in f around 0

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

5. Applied rewrites 96.5%

   \[\leadsto -\color{blue}{\mathsf{fma}\left(2 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi}, 0.125, -2 \cdot \frac{{\pi}^{3} \cdot 0.005208333333333333}{{\left(\pi \cdot 0.5\right)}^{2}}\right), \pi \cdot 0.5, {\left(\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right) \cdot \left(\pi \cdot 0.5\right)\right)}^{2} \cdot -0.25\right), f, \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right) \cdot \left(\pi \cdot 0.5\right)\right)}{\pi}, f, \frac{\log \left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)}{\pi} \cdot 4\right)} \]

6. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. distribute-rgt-out N/A

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

   4. lower-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. metadata-eval 96.5

      \[\leadsto -\mathsf{fma}\left(\left(\pi \cdot 0.08333333333333333\right) \cdot f, f, \frac{\log \left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)}{\pi} \cdot 4\right) \]

8. Applied rewrites 96.5%

   \[\leadsto -\mathsf{fma}\left(\left(\pi \cdot 0.08333333333333333\right) \cdot f, f, \frac{\log \left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)}{\pi} \cdot 4\right) \]

9. Taylor expanded in f around 0

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

    1. lower-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-PI.f64 96.5

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

11. Applied rewrites 96.5%

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

Alternative 3: 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. Add Preprocessing

3. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 96.1%

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

6. Taylor expanded in f around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-PI.f64 96.1

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

8. Applied rewrites 96.1%

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

Alternative 4: 4.2% accurate, 34.2× speedup

Math

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

FPCore

(FPCore (f) :precision binary64 (- (* (* (* f f) PI) 0.08333333333333333)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := (-N[(N[(N[(f * f), $MachinePrecision] * Pi), $MachinePrecision] * 0.08333333333333333), $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. Add Preprocessing

3. Taylor expanded in f around 0

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

5. Applied rewrites 96.5%

   \[\leadsto -\color{blue}{\mathsf{fma}\left(2 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi}, 0.125, -2 \cdot \frac{{\pi}^{3} \cdot 0.005208333333333333}{{\left(\pi \cdot 0.5\right)}^{2}}\right), \pi \cdot 0.5, {\left(\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right) \cdot \left(\pi \cdot 0.5\right)\right)}^{2} \cdot -0.25\right), f, \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right) \cdot \left(\pi \cdot 0.5\right)\right)}{\pi}, f, \frac{\log \left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)}{\pi} \cdot 4\right)} \]

6. Taylor expanded in f around inf

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

   1. *-commutative N/A

      \[\leadsto -\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right) \cdot {f}^{\color{blue}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto -\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right) \cdot {f}^{\color{blue}{2}} \]

   3. distribute-rgt-out N/A

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

   4. lower-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. metadata-eval N/A

      \[\leadsto -\left(\pi \cdot \frac{1}{12}\right) \cdot {f}^{2} \]

   7. unpow2 N/A

      \[\leadsto -\left(\pi \cdot \frac{1}{12}\right) \cdot \left(f \cdot f\right) \]

   8. lower-*.f64 4.2

      \[\leadsto -\left(\pi \cdot 0.08333333333333333\right) \cdot \left(f \cdot f\right) \]

8. Applied rewrites 4.2%

   \[\leadsto -\left(\pi \cdot 0.08333333333333333\right) \cdot \color{blue}{\left(f \cdot f\right)} \]

9. Taylor expanded in f around 0

   \[\leadsto -\frac{1}{12} \cdot \left({f}^{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto -\left({f}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{12} \]

    2. lower-*.f64 N/A

       \[\leadsto -\left({f}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{12} \]

    3. lower-*.f64 N/A

       \[\leadsto -\left({f}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{12} \]

    4. pow2 N/A

       \[\leadsto -\left(\left(f \cdot f\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{12} \]

    5. lift-*.f64 N/A

       \[\leadsto -\left(\left(f \cdot f\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{12} \]

    6. lift-PI.f64 4.2

       \[\leadsto -\left(\left(f \cdot f\right) \cdot \pi\right) \cdot 0.08333333333333333 \]

11. Applied rewrites 4.2%

    \[\leadsto -\left(\left(f \cdot f\right) \cdot \pi\right) \cdot 0.08333333333333333 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025091 
(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)))))))))