VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 98.9%

Time: 18.9s

Alternatives: 7

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
3. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

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-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. associate-*l/ N/A

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

   6. div-inv N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. associate-/l* N/A

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

   11. associate-*r* N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. clear-num N/A

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

   16. div-inv N/A

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

   17. lower-/.f64 N/A

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

   18. 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
3. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

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: 96.3% 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 4: 96.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(f \cdot \mathsf{fma}\left(f, f \cdot \left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -0.005208333333333333\right)\right), \pi \cdot 0.25\right)\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (log
   (* f (fma f (* f (* PI (* (* PI PI) -0.005208333333333333))) (* PI 0.25))))
  (* PI 0.25)))

C

double code(double f) {
	return log((f * fma(f, (f * (((double) M_PI) * ((((double) M_PI) * ((double) M_PI)) * -0.005208333333333333))), (((double) M_PI) * 0.25)))) / (((double) M_PI) * 0.25);
}

Julia

function code(f)
	return Float64(log(Float64(f * fma(f, Float64(f * Float64(pi * Float64(Float64(pi * pi) * -0.005208333333333333))), Float64(pi * 0.25)))) / Float64(pi * 0.25))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(f * N[(f * N[(Pi * N[(N[(Pi * Pi), $MachinePrecision] * -0.005208333333333333), $MachinePrecision]), $MachinePrecision]), $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
3. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

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-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. associate-*l/ N/A

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

   6. div-inv N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. associate-/l* N/A

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

   11. associate-*r* N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. clear-num N/A

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

   16. div-inv N/A

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

   17. lower-/.f64 N/A

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

   18. 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. Taylor expanded in f around 0

    \[\leadsto \frac{\log \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) + \color{blue}{{f}^{2} \cdot \left(\frac{-7}{768} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{1}{256} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]
11. Step-by-step derivation

12. Applied rewrites 97.1%

    \[\leadsto \frac{\log \left(f \cdot \mathsf{fma}\left(f, \color{blue}{f \cdot \left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -0.005208333333333333\right)\right)}, \pi \cdot 0.25\right)\right)}{\pi \cdot 0.25} \]
13. Add Preprocessing

Alternative 5: 95.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(N[Log[N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(f * 0.25), $MachinePrecision] * N[Sqrt[Pi], $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
3. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

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. Taylor expanded in f around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 96.5

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

7. Applied rewrites 96.5%

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

9. Applied rewrites 96.5%

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

10. Final simplification 96.5%

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

Alternative 6: 95.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(N[Log[N[(0.25 * N[(Pi * f), $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
3. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

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. Taylor expanded in f around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 96.5

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

7. Applied rewrites 96.5%

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

8. Final simplification 96.5%

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

Alternative 7: 95.8% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

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. Taylor expanded in f around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 96.5

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

7. Applied rewrites 96.5%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. rem-square-sqrt N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. metadata-eval N/A

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

   12. frac-times N/A

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

   13. lift-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift-*.f64 N/A

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

9. Applied rewrites 96.4%

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