VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.0% → 99.0%

Time: 14.2s

Alternatives: 7

Speedup: 4.6×

• could not determine a ground truth

  1. f = 1141640886.2232852

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: 7.0% 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: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-0.25 \cdot \left(\pi \cdot f\right)}\\ t_1 := 0.25 \cdot \left(\pi \cdot f\right)\\ \mathbf{if}\;f \leq 500:\\ \;\;\;\;-\frac{\log \left(\frac{\cosh t\_1}{\sinh t\_1}\right)}{\pi} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{1 + t\_0}{1 - t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (* -0.25 (* PI f)))) (t_1 (* 0.25 (* PI f))))
   (if (<= f 500.0)
     (- (* (/ (log (/ (cosh t_1) (sinh t_1))) PI) 4.0))
     (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ 1.0 t_0) (- 1.0 t_0))))))))

C

double code(double f) {
	double t_0 = exp((-0.25 * (((double) M_PI) * f)));
	double t_1 = 0.25 * (((double) M_PI) * f);
	double tmp;
	if (f <= 500.0) {
		tmp = -((log((cosh(t_1) / sinh(t_1))) / ((double) M_PI)) * 4.0);
	} else {
		tmp = -((1.0 / (((double) M_PI) / 4.0)) * log(((1.0 + t_0) / (1.0 - t_0))));
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = Math.exp((-0.25 * (Math.PI * f)));
	double t_1 = 0.25 * (Math.PI * f);
	double tmp;
	if (f <= 500.0) {
		tmp = -((Math.log((Math.cosh(t_1) / Math.sinh(t_1))) / Math.PI) * 4.0);
	} else {
		tmp = -((1.0 / (Math.PI / 4.0)) * Math.log(((1.0 + t_0) / (1.0 - t_0))));
	}
	return tmp;
}

Python

def code(f):
	t_0 = math.exp((-0.25 * (math.pi * f)))
	t_1 = 0.25 * (math.pi * f)
	tmp = 0
	if f <= 500.0:
		tmp = -((math.log((math.cosh(t_1) / math.sinh(t_1))) / math.pi) * 4.0)
	else:
		tmp = -((1.0 / (math.pi / 4.0)) * math.log(((1.0 + t_0) / (1.0 - t_0))))
	return tmp

Julia

function code(f)
	t_0 = exp(Float64(-0.25 * Float64(pi * f)))
	t_1 = Float64(0.25 * Float64(pi * f))
	tmp = 0.0
	if (f <= 500.0)
		tmp = Float64(-Float64(Float64(log(Float64(cosh(t_1) / sinh(t_1))) / pi) * 4.0));
	else
		tmp = Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(1.0 + t_0) / Float64(1.0 - t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	t_0 = exp((-0.25 * (pi * f)));
	t_1 = 0.25 * (pi * f);
	tmp = 0.0;
	if (f <= 500.0)
		tmp = -((log((cosh(t_1) / sinh(t_1))) / pi) * 4.0);
	else
		tmp = -((1.0 / (pi / 4.0)) * log(((1.0 + t_0) / (1.0 - t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := Block[{t$95$0 = N[Exp[N[(-0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.25 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[f, 500.0], (-N[(N[(N[Log[N[(N[Cosh[t$95$1], $MachinePrecision] / N[Sinh[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 4.0), $MachinePrecision]), (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(1.0 + t$95$0), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 2 regimes

2. if f < 500

   1. Initial program 7.1%

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

   3. Applied rewrites 99.0%

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in f around 0

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

   8. Applied rewrites 99.0%

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

   9. Taylor expanded in f around 0

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

   11. Applied rewrites 99.0%

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

   if 500 < f

   1. Initial program 4.3%

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

   2. Taylor expanded in f around 0

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

   4. Applied rewrites 1.6%

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

   5. Taylor expanded in f around 0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in f around 0

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

   10. Applied rewrites 100.0%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{1 + e^{\color{blue}{-0.25 \cdot \left(\pi \cdot f\right)}}}{1 - e^{-\frac{\pi}{4} \cdot f}}\right) \]

   11. Taylor expanded in f around 0

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

   13. Applied rewrites 100.0%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{1 + e^{-0.25 \cdot \left(\pi \cdot f\right)}}{1 - e^{\color{blue}{-0.25 \cdot \left(\pi \cdot f\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

3. Applied rewrites 97.1%

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

5. Applied rewrites 97.1%

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

6. Taylor expanded in f around 0

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

8. Applied rewrites 97.1%

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

9. Taylor expanded in f around 0

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

11. Applied rewrites 97.1%

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

Alternative 3: 96.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

3. Applied rewrites 97.1%

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

5. Applied rewrites 97.1%

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

6. Taylor expanded in f around 0

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

8. Applied rewrites 97.1%

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

9. Taylor expanded in f around 0

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

11. Applied rewrites 97.1%

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

12. Taylor expanded in f around 0

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

14. Applied rewrites 96.3%

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

Alternative 4: 96.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ -\frac{1 \cdot \log \left(\frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(f \cdot f\right), 0.0625, 2\right)}{\left(\pi \cdot \left(0.25 - -0.25\right)\right) \cdot f}\right)}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (/
   (*
    1.0
    (log
     (/ (fma (* (* PI PI) (* f f)) 0.0625 2.0) (* (* PI (- 0.25 -0.25)) f))))
   (/ PI 4.0))))

C

double code(double f) {
	return -((1.0 * log((fma(((((double) M_PI) * ((double) M_PI)) * (f * f)), 0.0625, 2.0) / ((((double) M_PI) * (0.25 - -0.25)) * f)))) / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(-Float64(Float64(1.0 * log(Float64(fma(Float64(Float64(pi * pi) * Float64(f * f)), 0.0625, 2.0) / Float64(Float64(pi * Float64(0.25 - -0.25)) * f)))) / Float64(pi / 4.0)))
end

Wolfram

code[f_] := (-N[(N[(1.0 * N[Log[N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(f * f), $MachinePrecision]), $MachinePrecision] * 0.0625 + 2.0), $MachinePrecision] / N[(N[(Pi * N[(0.25 - -0.25), $MachinePrecision]), $MachinePrecision] * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 7.0%

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

3. Applied rewrites 97.1%

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

4. Taylor expanded in f around 0

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

6. Applied rewrites 95.9%

   \[\leadsto -\frac{1 \cdot \log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{\color{blue}{\left(\pi \cdot \left(0.25 - -0.25\right)\right) \cdot f}}\right)}{\frac{\pi}{4}} \]

7. Taylor expanded in f around 0

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

9. Applied rewrites 96.0%

   \[\leadsto -\frac{1 \cdot \log \left(\frac{\color{blue}{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(f \cdot f\right), 0.0625, 2\right)}}{\left(\pi \cdot \left(0.25 - -0.25\right)\right) \cdot f}\right)}{\frac{\pi}{4}} \]
10. Add Preprocessing

Alternative 5: 95.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

3. Applied rewrites 97.1%

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

4. Taylor expanded in f around 0

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

6. Applied rewrites 95.9%

   \[\leadsto -\frac{1 \cdot \log \color{blue}{\left(\frac{2}{\left(\pi \cdot \left(0.25 - -0.25\right)\right) \cdot f}\right)}}{\frac{\pi}{4}} \]

7. Taylor expanded in f around 0

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

9. Applied rewrites 95.9%

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

Alternative 6: 95.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in f around 0

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

4. Applied rewrites 95.7%

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

5. Taylor expanded in f around 0

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

7. Applied rewrites 95.7%

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

9. Applied rewrites 95.7%

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

11. Applied rewrites 95.7%

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

Alternative 7: 95.7% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in f around 0

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

4. Applied rewrites 95.7%

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

5. Taylor expanded in f around 0

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

7. Applied rewrites 95.7%

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

9. Applied rewrites 95.7%

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

11. Applied rewrites 95.7%

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

Reproduce

herbie shell --seed 2025134 
(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)))))))))