VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 98.8%

Time: 10.5s

Alternatives: 6

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1014301221.906953

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot f\right) \cdot -0.25\\ t_1 := e^{t\_0}\\ \mathbf{if}\;f \leq 5:\\ \;\;\;\;\frac{\log \left(\frac{\cosh t\_0}{\sinh \left(\left(\pi \cdot f\right) \cdot 0.25\right)}\right) \cdot -4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{\pi} \cdot \left(-\log \left(\frac{1 + t\_1}{1 - t\_1}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (* PI f) -0.25)) (t_1 (exp t_0)))
   (if (<= f 5.0)
     (/ (* (log (/ (cosh t_0) (sinh (* (* PI f) 0.25)))) -4.0) PI)
     (* (/ 4.0 PI) (- (log (/ (+ 1.0 t_1) (- 1.0 t_1))))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) * f) * -0.25;
	double t_1 = exp(t_0);
	double tmp;
	if (f <= 5.0) {
		tmp = (log((cosh(t_0) / sinh(((((double) M_PI) * f) * 0.25)))) * -4.0) / ((double) M_PI);
	} else {
		tmp = (4.0 / ((double) M_PI)) * -log(((1.0 + t_1) / (1.0 - t_1)));
	}
	return tmp;
}

Java

public static double code(double f) {
	double t_0 = (Math.PI * f) * -0.25;
	double t_1 = Math.exp(t_0);
	double tmp;
	if (f <= 5.0) {
		tmp = (Math.log((Math.cosh(t_0) / Math.sinh(((Math.PI * f) * 0.25)))) * -4.0) / Math.PI;
	} else {
		tmp = (4.0 / Math.PI) * -Math.log(((1.0 + t_1) / (1.0 - t_1)));
	}
	return tmp;
}

Python

def code(f):
	t_0 = (math.pi * f) * -0.25
	t_1 = math.exp(t_0)
	tmp = 0
	if f <= 5.0:
		tmp = (math.log((math.cosh(t_0) / math.sinh(((math.pi * f) * 0.25)))) * -4.0) / math.pi
	else:
		tmp = (4.0 / math.pi) * -math.log(((1.0 + t_1) / (1.0 - t_1)))
	return tmp

Julia

function code(f)
	t_0 = Float64(Float64(pi * f) * -0.25)
	t_1 = exp(t_0)
	tmp = 0.0
	if (f <= 5.0)
		tmp = Float64(Float64(log(Float64(cosh(t_0) / sinh(Float64(Float64(pi * f) * 0.25)))) * -4.0) / pi);
	else
		tmp = Float64(Float64(4.0 / pi) * Float64(-log(Float64(Float64(1.0 + t_1) / Float64(1.0 - t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	t_0 = (pi * f) * -0.25;
	t_1 = exp(t_0);
	tmp = 0.0;
	if (f <= 5.0)
		tmp = (log((cosh(t_0) / sinh(((pi * f) * 0.25)))) * -4.0) / pi;
	else
		tmp = (4.0 / pi) * -log(((1.0 + t_1) / (1.0 - t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi * f), $MachinePrecision] * -0.25), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, If[LessEqual[f, 5.0], N[(N[(N[Log[N[(N[Cosh[t$95$0], $MachinePrecision] / N[Sinh[N[(N[(Pi * f), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -4.0), $MachinePrecision] / Pi), $MachinePrecision], N[(N[(4.0 / Pi), $MachinePrecision] * (-N[Log[N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if f < 5

   1. Initial program 6.9%

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

   2. Taylor expanded in f around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot f\right) \cdot -0.25\right)}{2 \cdot \sinh \left(\left(\pi \cdot f\right) \cdot 0.25\right)}\right)}{\pi} \cdot -4} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

   6. Applied rewrites 97.1%

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

   if 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

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

   4. Applied rewrites 5.3%

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

   5. Taylor expanded in f around 0

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

   7. Applied rewrites 6.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 6.2%

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

   10. Taylor expanded in f around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 N/A

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

       5. lift-PI.f64 6.2

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

   12. Applied rewrites 6.2%

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

   13. Taylor expanded in f around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 N/A

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

       5. lift-PI.f64 6.2

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

   15. Applied rewrites 6.2%

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

Alternative 2: 97.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 97.1%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot f\right) \cdot -0.25\right)}{2 \cdot \sinh \left(\left(\pi \cdot f\right) \cdot 0.25\right)}\right)}{\pi} \cdot -4} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

6. Applied rewrites 97.1%

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

Alternative 3: 96.2% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (/
  (*
   (log (/ (fma (* 0.03125 (* f f)) (* PI PI) 1.0) (sinh (* (* PI f) 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(((((double) M_PI) * f) * 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(pi * f) * 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[(Pi * f), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -4.0), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 97.1%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot f\right) \cdot -0.25\right)}{2 \cdot \sinh \left(\left(\pi \cdot f\right) \cdot 0.25\right)}\right)}{\pi} \cdot -4} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

6. Applied rewrites 97.1%

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

7. Taylor expanded in f around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-PI.f64 96.2

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

9. Applied rewrites 96.2%

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

Alternative 4: 96.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 97.1%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot f\right) \cdot -0.25\right)}{2 \cdot \sinh \left(\left(\pi \cdot f\right) \cdot 0.25\right)}\right)}{\pi} \cdot -4} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

6. Applied rewrites 97.1%

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

7. Taylor expanded in f around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-PI.f64 96.2

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

9. Applied rewrites 96.2%

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

    1. lift-PI.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. associate-/l* N/A

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

    5. lower-*.f64 N/A

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

11. Applied rewrites 96.1%

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

Alternative 5: 95.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 97.1%

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

5. Taylor expanded in f around 0

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

   1. lower-*.f64 N/A

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

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

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

   3. metadata-eval N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 95.9

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

7. Applied rewrites 95.9%

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

8. Taylor expanded in f around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-PI.f64 95.9

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

10. Applied rewrites 95.9%

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

Alternative 6: 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 6.9%

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

2. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 95.8%

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

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 95.8

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

7. Applied rewrites 95.8%

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

Reproduce

herbie shell --seed 2025142 
(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)))))))))