VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 97.0%

Time: 9.7s

Alternatives: 4

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1138860015.4842358

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: 97.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(N[(N[Log[N[(N[Cosh[N[(N[(f * Pi), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] / N[Sinh[N[(N[(0.25 * f), $MachinePrecision] * Pi), $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 97.0%

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

6. Applied rewrites 97.0%

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

Alternative 2: 96.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f \cdot f, \frac{4}{\pi}\right)}{f}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (*
   (/ 1.0 (/ PI 4.0))
   (log (/ (fma (* PI 0.08333333333333333) (* f f) (/ 4.0 PI)) f)))))

C

double code(double f) {
	return -((1.0 / (((double) M_PI) / 4.0)) * log((fma((((double) M_PI) * 0.08333333333333333), (f * f), (4.0 / ((double) M_PI))) / f)));
}

Julia

function code(f)
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(fma(Float64(pi * 0.08333333333333333), Float64(f * f), Float64(4.0 / pi)) / f))))
end

Wolfram

code[f_] := (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(N[(Pi * 0.08333333333333333), $MachinePrecision] * N[(f * f), $MachinePrecision] + N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $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 0

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

4. Applied rewrites 96.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(\frac{2}{\pi}, 2, \mathsf{fma}\left(\frac{\pi}{\pi \cdot 0.5}, 0, \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi}, 0.125, -2 \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333}{\left(\pi \cdot \pi\right) \cdot 0.25}\right) \cdot f\right) \cdot f\right)}{f}\right)} \]

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. distribute-rgt-out N/A

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

   9. lower-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. metadata-eval N/A

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\pi \cdot \frac{1}{12}, {f}^{2}, \frac{1}{\frac{\pi}{4}}\right)}{f}\right) \]

   12. unpow2 N/A

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

   13. lower-*.f64 96.2

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f \cdot f, \frac{1}{\frac{\pi}{4}}\right)}{f}\right) \]

   14. lift-/.f64 N/A

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

7. Applied rewrites 96.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f \cdot f, \frac{4}{\pi}\right)}{f}\right) \]
8. Add Preprocessing

Alternative 3: 95.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - (-(-N[Log[f], $MachinePrecision]))), $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 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 5.4%

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

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 6.4%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. log-div N/A

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

   4. lower--.f64 N/A

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

9. Applied rewrites 6.4%

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

10. Taylor expanded in f around 0

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

    1. lower--.f64 N/A

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

12. Applied rewrites 95.7%

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\mathsf{fma}\left(0 \cdot f, 0.5, \log 2\right) - \log \left(\left(0.5 \cdot \pi\right) \cdot f\right)\right)} \]

13. Taylor expanded in f around inf

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

    1. associate--r+ N/A

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

    2. *-commutative N/A

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

    3. metadata-eval N/A

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

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

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

    5. log-div N/A

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

    6. lower--.f64 N/A

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

15. Applied rewrites 95.8%

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

Alternative 4: 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 2025130 
(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)))))))))