VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.1%

Time: 26.6s

Alternatives: 7

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1474429493.1160161

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: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-0.25 \cdot \left(\pi \cdot f\right)}\\ t_1 := 0.25 \cdot \left(\pi \cdot f\right)\\ \mathbf{if}\;f \leq 23.5:\\ \;\;\;\;-\frac{\log \cosh t\_1 - \log \sinh t\_1}{\pi} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-\frac{1 \cdot 4}{\pi} \cdot \log \left(\frac{1 + t\_0}{1 - t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (* -0.25 (* PI f)))) (t_1 (* 0.25 (* PI f))))
   (if (<= f 23.5)
     (- (* (/ (- (log (cosh t_1)) (log (sinh t_1))) PI) 4.0))
     (- (* (/ (* 1.0 4.0) PI) (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 <= 23.5) {
		tmp = -(((log(cosh(t_1)) - log(sinh(t_1))) / ((double) M_PI)) * 4.0);
	} else {
		tmp = -(((1.0 * 4.0) / ((double) M_PI)) * 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 <= 23.5) {
		tmp = -(((Math.log(Math.cosh(t_1)) - Math.log(Math.sinh(t_1))) / Math.PI) * 4.0);
	} else {
		tmp = -(((1.0 * 4.0) / Math.PI) * 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 <= 23.5:
		tmp = -(((math.log(math.cosh(t_1)) - math.log(math.sinh(t_1))) / math.pi) * 4.0)
	else:
		tmp = -(((1.0 * 4.0) / math.pi) * 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 <= 23.5)
		tmp = Float64(-Float64(Float64(Float64(log(cosh(t_1)) - log(sinh(t_1))) / pi) * 4.0));
	else
		tmp = Float64(-Float64(Float64(Float64(1.0 * 4.0) / pi) * 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 <= 23.5)
		tmp = -(((log(cosh(t_1)) - log(sinh(t_1))) / pi) * 4.0);
	else
		tmp = -(((1.0 * 4.0) / pi) * 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, 23.5], (-N[(N[(N[(N[Log[N[Cosh[t$95$1], $MachinePrecision]], $MachinePrecision] - N[Log[N[Sinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * 4.0), $MachinePrecision]), (-N[(N[(N[(1.0 * 4.0), $MachinePrecision] / Pi), $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 < 23.5

   1. Initial program 6.7%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in f around 0

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

   10. Applied rewrites 99.4%

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

   11. Taylor expanded in f around 0

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

   13. Applied rewrites 99.4%

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

   if 23.5 < f

   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

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

   4. Applied rewrites 1.7%

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

   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 86.3%

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

   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 86.3%

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

   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 86.3%

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

   15. Applied rewrites 86.3%

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

Alternative 2: 97.2% accurate, 1.6× speedup

Math

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

FPCore

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

Julia

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

MATLAB

function tmp = code(f)
	t_0 = 0.25 * (pi * f);
	tmp = -(((log(cosh(t_0)) - log(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[(N[Log[N[Cosh[t$95$0], $MachinePrecision]], $MachinePrecision] - N[Log[N[Sinh[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * 4.0), $MachinePrecision])]

Derivation

1. Initial program 6.7%

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

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%

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

7. Applied rewrites 97.2%

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

8. Taylor expanded in f around 0

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

10. Applied rewrites 97.2%

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

11. Taylor expanded in f around 0

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

13. Applied rewrites 97.2%

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

Alternative 3: 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 6.7%

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

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 4: 96.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

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%

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

7. Applied rewrites 97.2%

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

8. Taylor expanded in f around 0

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

10. Applied rewrites 96.4%

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

Alternative 5: 96.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

double code(double f) {
	return -((1.0 * log((fma(0.0625, ((((double) M_PI) * ((double) M_PI)) * (f * f)), 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(0.0625, Float64(Float64(pi * pi) * Float64(f * f)), 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[(0.0625 * N[(N[(Pi * Pi), $MachinePrecision] * N[(f * f), $MachinePrecision]), $MachinePrecision] + 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 6.7%

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

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 96.0%

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

7. Taylor expanded in f around 0

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

9. Applied rewrites 96.0%

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

10. Taylor expanded in f around 0

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

12. Applied rewrites 96.0%

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

Alternative 6: 96.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

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

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

6. Applied rewrites 96.0%

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

7. Taylor expanded in f around inf

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

9. Applied rewrites 96.0%

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

Alternative 7: 95.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

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

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

6. Applied rewrites 96.0%

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

7. Taylor expanded in f around 0

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

9. Applied rewrites 95.9%

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

Reproduce

herbie shell --seed 2025121 
(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)))))))))