VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.1%

Time: 8.5s

Alternatives: 4

Speedup: 4.9×

• could not determine a ground truth

  1. f = 1100616206.450487

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, 3.5× speedup

Math

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

FPCore

(FPCore (f) :precision binary64 (* (/ (log (tanh (* (* f PI) 0.25))) PI) 4.0))

C

double code(double f) {
	return (log(tanh(((f * ((double) M_PI)) * 0.25))) / ((double) M_PI)) * 4.0;
}

Java

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

Python

def code(f):
	return (math.log(math.tanh(((f * math.pi) * 0.25))) / math.pi) * 4.0

Julia

function code(f)
	return Float64(Float64(log(tanh(Float64(Float64(f * pi) * 0.25))) / pi) * 4.0)
end

MATLAB

function tmp = code(f)
	tmp = (log(tanh(((f * pi) * 0.25))) / pi) * 4.0;
end

Wolfram

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

Derivation

1. Initial program 6.7%

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

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 97.2%

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

6. Applied rewrites 97.2%

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

7. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

9. Applied rewrites 99.1%

   \[\leadsto \frac{\log \tanh \left(\left(f \cdot \pi\right) \cdot 0.25\right)}{\pi} \cdot \color{blue}{4} \]
10. Add Preprocessing

Alternative 2: 96.2% accurate, 3.8× speedup

Math

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

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)) - 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%

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

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 96.1%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)}{\pi} \cdot -4} \]
5. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. metadata-eval N/A

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

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

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

   11. log-div N/A

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

6. Applied rewrites 96.2%

   \[\leadsto \frac{\log \left(\frac{2}{0.5 \cdot \pi}\right) - \log f}{\pi} \cdot -4 \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. lift-PI.f64 96.2

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

8. Applied rewrites 96.2%

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

Alternative 3: 96.1% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double f) {
	return (Math.log(((2.0 / f) / (0.5 * Math.PI))) / Math.PI) * -4.0;
}

Python

def code(f):
	return (math.log(((2.0 / f) / (0.5 * math.pi))) / math.pi) * -4.0

Julia

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

MATLAB

function tmp = code(f)
	tmp = (log(((2.0 / f) / (0.5 * pi))) / pi) * -4.0;
end

Wolfram

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

Derivation

1. Initial program 6.7%

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

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 96.1%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)}{\pi} \cdot -4} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. metadata-eval N/A

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

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

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-PI.f64 96.1

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

6. Applied rewrites 96.1%

   \[\leadsto \frac{\log \left(\frac{\frac{2}{f}}{0.5 \cdot \pi}\right)}{\pi} \cdot -4 \]
7. Add Preprocessing

Alternative 4: 96.1% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

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 97.2%

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

6. Applied rewrites 97.2%

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

7. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

9. Applied rewrites 99.1%

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

10. Taylor expanded in f around 0

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

    1. lower-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-PI.f64 96.1

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

12. Applied rewrites 96.1%

    \[\leadsto \frac{\log \left(0.25 \cdot \left(f \cdot \pi\right)\right)}{\pi} \cdot 4 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025115 
(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)))))))))