VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 99.0%

Time: 19.0s

Alternatives: 7

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1389004585.4416323

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: 99.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

4. Applied egg-rr 99.5%

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

   1. lift-PI.f64 99.5

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

   2. rem-square-sqrt N/A

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

   3. sqrt-unprod N/A

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

   4. lift-PI.f64 N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*r* N/A

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

   7. sqrt-prod N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-sqrt.f64 99.5

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

6. Applied egg-rr 99.5%

   \[\leadsto \frac{\log \tanh \left(\left(\color{blue}{\left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)} \cdot 0.25\right) \cdot f\right)}{\pi \cdot 0.25} \]
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 6.4%

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 98.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

4. Applied egg-rr 99.3%

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

5. Final simplification 99.3%

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

Alternative 4: 96.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(\pi \cdot \pi\right)\\ \frac{\log \left(f \cdot \mathsf{fma}\left(f, f \cdot \mathsf{fma}\left(t\_0, 0.00390625, t\_0 \cdot -0.009114583333333334\right), \pi \cdot 0.25\right)\right)}{\pi \cdot 0.25} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* PI (* PI PI))))
   (/
    (log
     (*
      f
      (fma
       f
       (* f (fma t_0 0.00390625 (* t_0 -0.009114583333333334)))
       (* PI 0.25))))
    (* PI 0.25))))

C

double code(double f) {
	double t_0 = ((double) M_PI) * (((double) M_PI) * ((double) M_PI));
	return log((f * fma(f, (f * fma(t_0, 0.00390625, (t_0 * -0.009114583333333334))), (((double) M_PI) * 0.25)))) / (((double) M_PI) * 0.25);
}

Julia

function code(f)
	t_0 = Float64(pi * Float64(pi * pi))
	return Float64(log(Float64(f * fma(f, Float64(f * fma(t_0, 0.00390625, Float64(t_0 * -0.009114583333333334))), Float64(pi * 0.25)))) / Float64(pi * 0.25))
end

Wolfram

code[f_] := Block[{t$95$0 = N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[Log[N[(f * N[(f * N[(f * N[(t$95$0 * 0.00390625 + N[(t$95$0 * -0.009114583333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.4%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in f around 0

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

7. Simplified 97.1%

   \[\leadsto \frac{1}{\frac{\pi \cdot 0.25}{\log \color{blue}{\left(f \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.03125, -0.125, \mathsf{fma}\left(\pi \cdot \left(\pi \cdot \pi\right), -0.0013020833333333333, -0.0078125 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right), f \cdot f, 0.25 \cdot \pi\right)\right)}}} \]

9. Applied egg-rr 97.2%

   \[\leadsto \color{blue}{\frac{\log \left(f \cdot \mathsf{fma}\left(f, f \cdot \mathsf{fma}\left(\pi \cdot \left(\pi \cdot \pi\right), 0.00390625, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.009114583333333334\right), \pi \cdot 0.25\right)\right)}{\pi \cdot 0.25}} \]
10. Add Preprocessing

Alternative 5: 95.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(\pi \cdot -0.25\right) \cdot 16\right) \cdot \frac{\log \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{-\pi \cdot \pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (* (* PI -0.25) 16.0) (/ (log (* PI (* 0.25 f))) (- (* PI PI)))))

C

double code(double f) {
	return ((((double) M_PI) * -0.25) * 16.0) * (log((((double) M_PI) * (0.25 * f))) / -(((double) M_PI) * ((double) M_PI)));
}

Java

public static double code(double f) {
	return ((Math.PI * -0.25) * 16.0) * (Math.log((Math.PI * (0.25 * f))) / -(Math.PI * Math.PI));
}

Python

def code(f):
	return ((math.pi * -0.25) * 16.0) * (math.log((math.pi * (0.25 * f))) / -(math.pi * math.pi))

Julia

function code(f)
	return Float64(Float64(Float64(pi * -0.25) * 16.0) * Float64(log(Float64(pi * Float64(0.25 * f))) / Float64(-Float64(pi * pi))))
end

MATLAB

function tmp = code(f)
	tmp = ((pi * -0.25) * 16.0) * (log((pi * (0.25 * f))) / -(pi * pi));
end

Wolfram

code[f_] := N[(N[(N[(Pi * -0.25), $MachinePrecision] * 16.0), $MachinePrecision] * N[(N[Log[N[(Pi * N[(0.25 * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(Pi * Pi), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.4%

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

4. Applied egg-rr 99.5%

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

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. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 96.6

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

7. Simplified 96.6%

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

9. Applied egg-rr 96.4%

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

    1. lift-PI.f64 N/A

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

    2. lift-PI.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-log.f64 N/A

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

    6. associate-*l/ N/A

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

    7. *-commutative N/A

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

    8. metadata-eval N/A

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

    9. div-inv N/A

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

    10. associate-/r* N/A

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

    11. metadata-eval N/A

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

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

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

    13. *-commutative N/A

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

    14. lift-*.f64 N/A

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

    15. distribute-frac-neg2 N/A

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

    16. distribute-frac-neg N/A

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

11. Applied egg-rr 96.6%

    \[\leadsto \color{blue}{\frac{\log \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\pi \cdot \pi} \cdot \left(-\left(\pi \cdot -0.25\right) \cdot 16\right)} \]

12. Final simplification 96.6%

    \[\leadsto \left(\left(\pi \cdot -0.25\right) \cdot 16\right) \cdot \frac{\log \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{-\pi \cdot \pi} \]
13. Add Preprocessing

Alternative 6: 95.8% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

4. Applied egg-rr 99.5%

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

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. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 96.6

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

7. Simplified 96.6%

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

8. Final simplification 96.6%

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

Alternative 7: 95.6% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

4. Applied egg-rr 99.5%

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

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. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 96.6

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

7. Simplified 96.6%

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

9. Applied egg-rr 96.4%

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

Reproduce

herbie shell --seed 2024208 
(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)))))))))