VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 98.9%

Time: 20.1s

Alternatives: 8

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1483174176.1859756

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: 6.8% 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: 98.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\log \tanh \left(\frac{\pi \cdot 0.5}{\frac{2}{f}}\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (log (tanh (/ (* PI 0.5) (/ 2.0 f)))) (* PI 0.25)))

C

double code(double f) {
	return log(tanh(((((double) M_PI) * 0.5) / (2.0 / f)))) / (((double) M_PI) * 0.25);
}

Java

public static double code(double f) {
	return Math.log(Math.tanh(((Math.PI * 0.5) / (2.0 / f)))) / (Math.PI * 0.25);
}

Python

def code(f):
	return math.log(math.tanh(((math.pi * 0.5) / (2.0 / f)))) / (math.pi * 0.25)

Julia

function code(f)
	return Float64(log(tanh(Float64(Float64(pi * 0.5) / Float64(2.0 / f)))) / Float64(pi * 0.25))
end

MATLAB

function tmp = code(f)
	tmp = log(tanh(((pi * 0.5) / (2.0 / f)))) / (pi * 0.25);
end

Wolfram

code[f_] := N[(N[Log[N[Tanh[N[(N[(Pi * 0.5), $MachinePrecision] / N[(2.0 / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

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

4. Applied rewrites 99.3%

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

   1. lift-PI.f64 N/A

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

   2. metadata-eval N/A

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

   3. div-inv N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. associate-/l* N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

   14. clear-num N/A

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

   15. div-inv N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 99.3

       \[\leadsto \frac{\log \tanh \left(\frac{\pi \cdot 0.5}{\color{blue}{\frac{2}{f}}}\right)}{\pi \cdot 0.25} \]

6. Applied rewrites 99.3%

   \[\leadsto \frac{\log \tanh \color{blue}{\left(\frac{\pi \cdot 0.5}{\frac{2}{f}}\right)}}{\pi \cdot 0.25} \]
7. Add Preprocessing

Alternative 2: 99.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.3%

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

4. Applied rewrites 99.3%

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

5. Final simplification 99.3%

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

Alternative 3: 96.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \log \left(\frac{f}{\mathsf{fma}\left(f \cdot \left(f \cdot -16\right), \frac{\pi \cdot \mathsf{fma}\left(\pi \cdot \pi, 0.00390625, \left(\pi \cdot \pi\right) \cdot -0.009114583333333334\right)}{\pi \cdot \pi}, \frac{4}{\pi}\right)}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (*
   4.0
   (log
    (/
     f
     (fma
      (* f (* f -16.0))
      (/
       (* PI (fma (* PI PI) 0.00390625 (* (* PI PI) -0.009114583333333334)))
       (* PI PI))
      (/ 4.0 PI)))))
  PI))

C

double code(double f) {
	return (4.0 * log((f / fma((f * (f * -16.0)), ((((double) M_PI) * fma((((double) M_PI) * ((double) M_PI)), 0.00390625, ((((double) M_PI) * ((double) M_PI)) * -0.009114583333333334))) / (((double) M_PI) * ((double) M_PI))), (4.0 / ((double) M_PI)))))) / ((double) M_PI);
}

Julia

function code(f)
	return Float64(Float64(4.0 * log(Float64(f / fma(Float64(f * Float64(f * -16.0)), Float64(Float64(pi * fma(Float64(pi * pi), 0.00390625, Float64(Float64(pi * pi) * -0.009114583333333334))) / Float64(pi * pi)), Float64(4.0 / pi))))) / pi)
end

Wolfram

code[f_] := N[(N[(4.0 * N[Log[N[(f / N[(N[(f * N[(f * -16.0), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * N[(N[(Pi * Pi), $MachinePrecision] * 0.00390625 + N[(N[(Pi * Pi), $MachinePrecision] * -0.009114583333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 7.3%

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

4. Applied rewrites 99.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{1}{\tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}\right)} \]
5. Step-by-step derivation

   1. lift-PI.f64 N/A

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

   2. clear-num N/A

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

   3. lift-PI.f64 N/A

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

   4. add-sqr-sqrt N/A

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

   5. associate-/r* N/A

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

   6. div-inv N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-sqrt.f64 98.6

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

6. Applied rewrites 98.6%

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

7. Taylor expanded in f around 0

   \[\leadsto \mathsf{neg}\left(\left(\frac{4}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \log \color{blue}{\left(\frac{f \cdot \left(-1 \cdot \left(f \cdot \left(-16 \cdot \frac{{\left(\frac{-1}{16} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{16} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}^{2}}{{\mathsf{PI}\left(\right)}^{3}} + 16 \cdot \frac{\frac{1}{2} \cdot \left(\frac{-1}{4} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{16} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{32} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \left(\frac{-1}{128} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{1}{192} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) - \frac{1}{128} \cdot {\mathsf{PI}\left(\right)}^{3}}{{\mathsf{PI}\left(\right)}^{2}}\right)\right) - 8 \cdot \frac{\frac{-1}{16} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{16} \cdot {\mathsf{PI}\left(\right)}^{2}}{{\mathsf{PI}\left(\right)}^{2}}\right) + 4 \cdot \frac{1}{\mathsf{PI}\left(\right)}}{f}\right)}\right) \]

9. Applied rewrites 96.5%

   \[\leadsto -\left(\frac{4}{\sqrt{\pi}} \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(f, f \cdot \left(-16 \cdot \frac{\mathsf{fma}\left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -0.03125\right), -0.125, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.009114583333333334\right)}{\pi \cdot \pi}\right), \frac{4}{\pi}\right)}{f}\right)} \]

11. Applied rewrites 97.2%

    \[\leadsto \color{blue}{\frac{4 \cdot \log \left(\frac{f}{\mathsf{fma}\left(f \cdot \left(f \cdot -16\right), \frac{\pi \cdot \mathsf{fma}\left(\pi \cdot \pi, 0.00390625, \left(\pi \cdot \pi\right) \cdot -0.009114583333333334\right)}{\pi \cdot \pi}, \frac{4}{\pi}\right)}\right)}{\pi}} \]
12. Add Preprocessing

Alternative 4: 96.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{\mathsf{fma}\left(f, f \cdot \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \pi \cdot 0.125\right), \frac{4}{\pi}\right)}{f}\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    (fma
     f
     (* f (fma (* PI 0.020833333333333332) -2.0 (* PI 0.125)))
     (/ 4.0 PI))
    f))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log((fma(f, (f * fma((((double) M_PI) * 0.020833333333333332), -2.0, (((double) M_PI) * 0.125))), (4.0 / ((double) M_PI))) / f)) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(Float64(fma(f, Float64(f * fma(Float64(pi * 0.020833333333333332), -2.0, Float64(pi * 0.125))), Float64(4.0 / pi)) / f)) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(f * N[(f * N[(N[(Pi * 0.020833333333333332), $MachinePrecision] * -2.0 + N[(Pi * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

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

3. Taylor expanded in f around 0

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

5. Applied rewrites 97.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(f, f \cdot \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \pi \cdot 0.125\right), \frac{4}{\pi}\right)}{f}\right)} \]

6. Final simplification 97.1%

   \[\leadsto \log \left(\frac{\mathsf{fma}\left(f, f \cdot \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \pi \cdot 0.125\right), \frac{4}{\pi}\right)}{f}\right) \cdot \frac{-1}{\frac{\pi}{4}} \]
7. Add Preprocessing

Alternative 5: 96.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(f \cdot \mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -0.03125\right), -0.125, -0.009114583333333334 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), f \cdot f, \pi \cdot 0.25\right)\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (log
   (*
    f
    (fma
     (fma
      (* PI (* (* PI PI) -0.03125))
      -0.125
      (* -0.009114583333333334 (* PI (* PI PI))))
     (* f f)
     (* PI 0.25))))
  (* PI 0.25)))

C

double code(double f) {
	return log((f * fma(fma((((double) M_PI) * ((((double) M_PI) * ((double) M_PI)) * -0.03125)), -0.125, (-0.009114583333333334 * (((double) M_PI) * (((double) M_PI) * ((double) M_PI))))), (f * f), (((double) M_PI) * 0.25)))) / (((double) M_PI) * 0.25);
}

Julia

function code(f)
	return Float64(log(Float64(f * fma(fma(Float64(pi * Float64(Float64(pi * pi) * -0.03125)), -0.125, Float64(-0.009114583333333334 * Float64(pi * Float64(pi * pi)))), Float64(f * f), Float64(pi * 0.25)))) / Float64(pi * 0.25))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(N[(N[(Pi * N[(N[(Pi * Pi), $MachinePrecision] * -0.03125), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(-0.009114583333333334 * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(f * f), $MachinePrecision] + N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

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

4. Applied rewrites 99.3%

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

   1. lift-PI.f64 N/A

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

   2. metadata-eval N/A

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

   3. div-inv N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. associate-/l* N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

   14. clear-num N/A

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

   15. div-inv N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 99.3

       \[\leadsto \frac{\log \tanh \left(\frac{\pi \cdot 0.5}{\color{blue}{\frac{2}{f}}}\right)}{\pi \cdot 0.25} \]

6. Applied rewrites 99.3%

   \[\leadsto \frac{\log \tanh \color{blue}{\left(\frac{\pi \cdot 0.5}{\frac{2}{f}}\right)}}{\pi \cdot 0.25} \]

7. Taylor expanded in f around 0

   \[\leadsto \frac{\log \color{blue}{\left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) + f \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{16} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{16} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + f \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{4} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{16} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{32} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \left(\frac{-1}{128} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{1}{192} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) - \frac{1}{128} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)}}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

9. Applied rewrites 97.1%

   \[\leadsto \frac{\log \color{blue}{\left(f \cdot \mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -0.03125\right), -0.125, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.009114583333333334\right), f \cdot f, 0.25 \cdot \pi\right)\right)}}{\pi \cdot 0.25} \]

10. Final simplification 97.1%

    \[\leadsto \frac{\log \left(f \cdot \mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -0.03125\right), -0.125, -0.009114583333333334 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), f \cdot f, \pi \cdot 0.25\right)\right)}{\pi \cdot 0.25} \]
11. Add Preprocessing

Alternative 6: 95.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.3%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
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. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. metadata-eval 96.4

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

5. Applied rewrites 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. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.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. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-log.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. clear-num N/A

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

   14. lower-/.f64 N/A

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

   15. lift-log.f64 N/A

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

7. Applied rewrites 96.4%

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

   1. lift-PI.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. /-rgt-identity N/A

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

   4. clear-num N/A

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

   5. div-inv N/A

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

   6. log-prod N/A

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

   7. div-inv N/A

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

   8. clear-num N/A

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

   9. /-rgt-identity N/A

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

   10. lift-*.f64 N/A

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

   11. log-prod N/A

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

   12. lift-*.f64 N/A

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

   13. lift-log.f64 N/A

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

   14. associate-*l/ N/A

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

   15. lower-/.f64 N/A

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

9. Applied rewrites 96.5%

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

    1. lift-PI.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. rem-square-sqrt N/A

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

    5. lift-sqrt.f64 N/A

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

    6. lift-sqrt.f64 N/A

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

    7. associate-*r* N/A

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

    8. lower-*.f64 N/A

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

    9. lower-*.f64 96.5

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

    10. lift-*.f64 N/A

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

    11. *-commutative N/A

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

    12. lower-*.f64 96.5

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

11. Applied rewrites 96.5%

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

12. Final simplification 96.5%

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

Alternative 7: 95.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.3%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
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. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. metadata-eval 96.4

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

5. Applied rewrites 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. associate-/r/ N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. associate-*l* N/A

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

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

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

   10. distribute-rgt-neg-out N/A

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

   11. associate-*l* N/A

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

   12. associate-/r/ N/A

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

   13. clear-num N/A

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

7. Applied rewrites 96.5%

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

Alternative 8: 95.8% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.3%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
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. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. metadata-eval 96.4

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

5. Applied rewrites 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. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.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. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-log.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. clear-num N/A

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

   14. lower-/.f64 N/A

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

   15. lift-log.f64 N/A

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

7. Applied rewrites 96.4%

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

Reproduce

herbie shell --seed 2024221 
(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)))))))))