VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.1%

Time: 19.4s

Alternatives: 5

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1539514006.0073235

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: 6.7% 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: 99.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

4. Applied rewrites 98.7%

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

Alternative 2: 96.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[f_] := N[(N[Log[N[(f * N[(0.25 * Pi + N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(-0.005208333333333333 * N[(f * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in f around 0

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in f around 0

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

   1. lower-fma.f64 N/A

      \[\leadsto \frac{\log \left(f \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right), {f}^{2} \cdot \left(\frac{-7}{768} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{1}{256} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   2. lower-PI.f64 N/A

      \[\leadsto \frac{\log \left(f \cdot \mathsf{fma}\left(\frac{1}{4}, \color{blue}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-7}{768} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{1}{256} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\log \left(f \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right), \color{blue}{\left(\frac{-7}{768} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{1}{256} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {f}^{2}}\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   4. distribute-rgt-out N/A

      \[\leadsto \frac{\log \left(f \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\frac{-7}{768} + \frac{1}{256}\right)\right)} \cdot {f}^{2}\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\log \left(f \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right), \color{blue}{{\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(\frac{-7}{768} + \frac{1}{256}\right) \cdot {f}^{2}\right)}\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\log \left(f \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right), \color{blue}{{\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(\frac{-7}{768} + \frac{1}{256}\right) \cdot {f}^{2}\right)}\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   7. cube-mult N/A

      \[\leadsto \frac{\log \left(f \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(\left(\frac{-7}{768} + \frac{1}{256}\right) \cdot {f}^{2}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\log \left(f \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right) \cdot \left(\left(\frac{-7}{768} + \frac{1}{256}\right) \cdot {f}^{2}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\log \left(f \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \cdot \left(\left(\frac{-7}{768} + \frac{1}{256}\right) \cdot {f}^{2}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   10. lower-PI.f64 N/A

       \[\leadsto \frac{\log \left(f \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(\left(\frac{-7}{768} + \frac{1}{256}\right) \cdot {f}^{2}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\log \left(f \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left(\left(\frac{-7}{768} + \frac{1}{256}\right) \cdot {f}^{2}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\log \left(f \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left(\left(\frac{-7}{768} + \frac{1}{256}\right) \cdot {f}^{2}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   13. lower-PI.f64 N/A

       \[\leadsto \frac{\log \left(f \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(\frac{-7}{768} + \frac{1}{256}\right) \cdot {f}^{2}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   14. lower-PI.f64 N/A

       \[\leadsto \frac{\log \left(f \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\left(\frac{-7}{768} + \frac{1}{256}\right) \cdot {f}^{2}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\log \left(f \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(\frac{-7}{768} + \frac{1}{256}\right) \cdot {f}^{2}\right)}\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\log \left(f \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\frac{-1}{192}} \cdot {f}^{2}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   17. unpow2 N/A

       \[\leadsto \frac{\log \left(f \cdot \mathsf{fma}\left(\frac{1}{4}, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{-1}{192} \cdot \color{blue}{\left(f \cdot f\right)}\right)\right)\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{4}} \]

   18. lower-*.f64 97.7

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

10. Applied rewrites 97.7%

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

Alternative 3: 96.1% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

3. Taylor expanded in f around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 96.9

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

5. Applied rewrites 96.9%

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

7. Applied rewrites 97.4%

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

   1. lift-PI.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. metadata-eval N/A

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

   7. associate-/r/ N/A

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

   8. clear-num N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. un-div-inv N/A

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

   12. lower-/.f64 97.4

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

   13. lift-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. clear-num N/A

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

   16. lower-/.f64 97.4

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

9. Applied rewrites 97.4%

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

Alternative 4: 96.1% 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 6.5%

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

3. Taylor expanded in f around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 96.9

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

5. Applied rewrites 96.9%

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

7. Applied rewrites 97.4%

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

Alternative 5: 95.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

3. Taylor expanded in f around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 96.9

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

5. Applied rewrites 96.9%

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

7. Applied rewrites 97.3%

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

Reproduce

herbie shell --seed 2024216 
(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)))))))))