VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 99.0%

Time: 19.0s

Alternatives: 6

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1227118774.093243

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: 6.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t\_0}\\ t_2 := e^{-t\_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t\_1 + t\_2}{t\_1 - t\_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Alternative 1: 99.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{4}{\sqrt{\pi}} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (/ 4.0 (sqrt PI)) (* (/ 1.0 (sqrt PI)) (log (tanh (* f (* PI 0.25)))))))

C

double code(double f) {
	return (4.0 / sqrt(((double) M_PI))) * ((1.0 / sqrt(((double) M_PI))) * log(tanh((f * (((double) M_PI) * 0.25)))));
}

Java

public static double code(double f) {
	return (4.0 / Math.sqrt(Math.PI)) * ((1.0 / Math.sqrt(Math.PI)) * Math.log(Math.tanh((f * (Math.PI * 0.25)))));
}

Python

def code(f):
	return (4.0 / math.sqrt(math.pi)) * ((1.0 / math.sqrt(math.pi)) * math.log(math.tanh((f * (math.pi * 0.25)))))

Julia

function code(f)
	return Float64(Float64(4.0 / sqrt(pi)) * Float64(Float64(1.0 / sqrt(pi)) * log(tanh(Float64(f * Float64(pi * 0.25))))))
end

MATLAB

function tmp = code(f)
	tmp = (4.0 / sqrt(pi)) * ((1.0 / sqrt(pi)) * log(tanh((f * (pi * 0.25)))));
end

Wolfram

code[f_] := N[(N[(4.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Log[N[Tanh[N[(f * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.6%

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

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

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

   2. inv-pow N/A

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

   3. sqr-pow N/A

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

   4. associate-*l* N/A

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

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

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

4. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{{\left(\pi \cdot 0.25\right)}^{-0.5} \cdot \left({\left(\pi \cdot 0.25\right)}^{-0.5} \cdot \log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)\right)} \]
5. Step-by-step derivation

   1. associate-*r* N/A

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

   2. pow-prod-up N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. inv-pow N/A

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

   7. *-commutative N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. div-inv N/A

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

   11. *-commutative N/A

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

   12. associate-/r* N/A

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

   13. clear-num N/A

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

   14. unpow-1 N/A

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

   15. /-lowering-/.f64 N/A

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

6. Applied egg-rr 99.0%

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

   1. metadata-eval N/A

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

   2. div-inv N/A

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

   3. associate-/r* N/A

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

   4. add-sqr-sqrt N/A

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

   5. associate-*l* N/A

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

   6. div-inv N/A

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. div-inv N/A

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

   10. associate-*l* N/A

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

   11. *-lowering-*.f64 N/A

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

8. Applied egg-rr 99.0%

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

9. Final simplification 99.0%

   \[\leadsto \frac{4}{\sqrt{\pi}} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \log \tanh \left(f \cdot \left(\pi \cdot 0.25\right)\right)\right) \]
10. 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 6.6%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. distribute-neg-frac N/A

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

   4. /-lowering-/.f64 N/A

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

4. Applied egg-rr 99.0%

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

5. Final simplification 99.0%

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

Alternative 3: 96.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{\mathsf{fma}\left(f, f \cdot \mathsf{fma}\left(0.0625, \pi + \pi, \left(0.005208333333333333 \cdot \left(\left(\pi + \pi\right) \cdot 2\right)\right) \cdot -2\right), \frac{4}{\pi}\right)}{f}\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    (fma
     f
     (*
      f
      (fma
       0.0625
       (+ PI PI)
       (* (* 0.005208333333333333 (* (+ PI PI) 2.0)) -2.0)))
     (/ 4.0 PI))
    f))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log((fma(f, (f * fma(0.0625, (((double) M_PI) + ((double) M_PI)), ((0.005208333333333333 * ((((double) M_PI) + ((double) M_PI)) * 2.0)) * -2.0))), (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(0.0625, Float64(pi + pi), Float64(Float64(0.005208333333333333 * Float64(Float64(pi + pi) * 2.0)) * -2.0))), 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[(0.0625 * N[(Pi + Pi), $MachinePrecision] + N[(N[(0.005208333333333333 * N[(N[(Pi + Pi), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * -2.0), $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 6.6%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
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. Simplified 96.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(f, f \cdot \mathsf{fma}\left(0.0625, \pi + \pi, \left(0.005208333333333333 \cdot \left(2 \cdot \left(\pi + \pi\right)\right)\right) \cdot -2\right), \frac{4}{\pi}\right)}{f}\right)} \]

6. Final simplification 96.9%

   \[\leadsto \log \left(\frac{\mathsf{fma}\left(f, f \cdot \mathsf{fma}\left(0.0625, \pi + \pi, \left(0.005208333333333333 \cdot \left(\left(\pi + \pi\right) \cdot 2\right)\right) \cdot -2\right), \frac{4}{\pi}\right)}{f}\right) \cdot \frac{-1}{\frac{\pi}{4}} \]
7. Add Preprocessing

Alternative 4: 96.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \log \left(f \cdot \mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -0.03125\right), -0.125, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.009114583333333334\right), f \cdot f, \pi \cdot 0.25\right)\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (*
   4.0
   (log
    (*
     f
     (fma
      (fma
       (* PI (* (* PI PI) -0.03125))
       -0.125
       (* (* PI (* PI PI)) -0.009114583333333334))
      (* f f)
      (* PI 0.25)))))
  PI))

C

double code(double f) {
	return (4.0 * log((f * fma(fma((((double) M_PI) * ((((double) M_PI) * ((double) M_PI)) * -0.03125)), -0.125, ((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * -0.009114583333333334)), (f * f), (((double) M_PI) * 0.25))))) / ((double) M_PI);
}

Julia

function code(f)
	return Float64(Float64(4.0 * log(Float64(f * fma(fma(Float64(pi * Float64(Float64(pi * pi) * -0.03125)), -0.125, Float64(Float64(pi * Float64(pi * pi)) * -0.009114583333333334)), Float64(f * f), Float64(pi * 0.25))))) / pi)
end

Wolfram

code[f_] := N[(N[(4.0 * N[Log[N[(f * N[(N[(N[(Pi * N[(N[(Pi * Pi), $MachinePrecision] * -0.03125), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * -0.009114583333333334), $MachinePrecision]), $MachinePrecision] * N[(f * f), $MachinePrecision] + N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 6.6%

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

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

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

   2. inv-pow N/A

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

   3. sqr-pow N/A

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

   4. associate-*l* N/A

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

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

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

4. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{{\left(\pi \cdot 0.25\right)}^{-0.5} \cdot \left({\left(\pi \cdot 0.25\right)}^{-0.5} \cdot \log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)\right)} \]
5. Step-by-step derivation

   1. associate-*r* N/A

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

   2. pow-prod-up N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. inv-pow N/A

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

   7. *-commutative N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. div-inv N/A

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

   11. *-commutative N/A

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

   12. associate-/r* N/A

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

   13. clear-num N/A

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

   14. unpow-1 N/A

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

   15. /-lowering-/.f64 N/A

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in f around 0

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

9. Simplified 96.8%

   \[\leadsto \frac{\log \color{blue}{\left(f \cdot \mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -0.03125\right), -0.125, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.009114583333333334\right), f \cdot f, 0.25 \cdot \pi\right)\right)} \cdot 4}{\pi} \]

10. Final simplification 96.8%

    \[\leadsto \frac{4 \cdot \log \left(f \cdot \mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -0.03125\right), -0.125, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.009114583333333334\right), f \cdot f, \pi \cdot 0.25\right)\right)}{\pi} \]
11. Add Preprocessing

Alternative 5: 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 6.6%

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

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

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

   2. inv-pow N/A

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

   3. sqr-pow N/A

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

   4. associate-*l* N/A

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

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

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

4. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{{\left(\pi \cdot 0.25\right)}^{-0.5} \cdot \left({\left(\pi \cdot 0.25\right)}^{-0.5} \cdot \log \tanh \left(\left(\pi \cdot 0.25\right) \cdot f\right)\right)} \]
5. Step-by-step derivation

   1. associate-*r* N/A

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

   2. pow-prod-up N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. inv-pow N/A

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

   7. *-commutative N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. div-inv N/A

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

   11. *-commutative N/A

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

   12. associate-/r* N/A

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

   13. clear-num N/A

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

   14. unpow-1 N/A

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

   15. /-lowering-/.f64 N/A

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in f around 0

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

   1. *-lowering-*.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. PI-lowering-PI.f64 96.6

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

9. Simplified 96.6%

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

10. Final simplification 96.6%

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

Alternative 6: 95.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

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

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

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

   4. metadata-eval N/A

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

   5. *-lowering-*.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. PI-lowering-PI.f64 96.5

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

5. Simplified 96.5%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. un-div-inv N/A

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

   5. frac-2neg N/A

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

   6. neg-log N/A

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

   7. clear-num N/A

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

   8. /-lowering-/.f64 N/A

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

7. Applied egg-rr 96.6%

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

   1. distribute-neg-frac N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. log-rec N/A

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

   8. clear-num N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. distribute-lft-neg-in N/A

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

   12. add-sqr-sqrt N/A

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

   13. associate-*l* N/A

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

   14. distribute-frac-neg2 N/A

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

9. Applied egg-rr 96.5%

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

10. Final simplification 96.5%

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

Reproduce

herbie shell --seed 2024198 
(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)))))))))