VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.5% → 99.2%

Time: 17.6s

Alternatives: 4

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1104587935.7622051

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: 6.5% 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: 99.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.0%

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

4. Applied egg-rr 98.9%

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

5. Final simplification 98.9%

   \[\leadsto \frac{\log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{-\pi \cdot -0.25} \]
6. Add Preprocessing

Alternative 2: 96.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.0%

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

3. Taylor expanded in f around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 96.4

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

5. Simplified 96.4%

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

7. Applied egg-rr 96.4%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 96.5

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 96.5

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

9. Applied egg-rr 96.5%

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

    1. lift-PI.f64 N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

    4. lift-*.f64 N/A

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

    5. rem-square-sqrt N/A

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

    6. lift-sqrt.f64 N/A

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

    7. lift-sqrt.f64 N/A

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

    8. associate-*r* N/A

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

    9. lower-*.f64 N/A

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

    10. lower-*.f64 96.5

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

11. Applied egg-rr 96.5%

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

12. Final simplification 96.5%

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

Alternative 3: 96.1% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.0%

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

3. Taylor expanded in f around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 96.4

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

5. Simplified 96.4%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right)} \]
6. Step-by-step derivation

   1. lift-PI.f64 N/A

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

   2. frac-2neg N/A

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

   3. frac-2neg N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. associate-*l/ N/A

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

7. Applied egg-rr 96.5%

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

Alternative 4: 95.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.0%

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

3. Taylor expanded in f around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 96.4

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

5. Simplified 96.4%

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

7. Applied egg-rr 96.4%

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

8. Final simplification 96.4%

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

Reproduce

herbie shell --seed 2024219 
(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)))))))))