VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 99.0%

Time: 33.7s

Alternatives: 5

Speedup: 4.9×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

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, 1.7× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot -0.5\right)}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log
    (+ (/ 1.0 (expm1 (* (* f PI) 0.5))) (/ -1.0 (expm1 (* (* f PI) -0.5)))))
   PI)))

C

double code(double f) {
	return -4.0 * (log(((1.0 / expm1(((f * ((double) M_PI)) * 0.5))) + (-1.0 / expm1(((f * ((double) M_PI)) * -0.5))))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log(((1.0 / Math.expm1(((f * Math.PI) * 0.5))) + (-1.0 / Math.expm1(((f * Math.PI) * -0.5))))) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log(((1.0 / math.expm1(((f * math.pi) * 0.5))) + (-1.0 / math.expm1(((f * math.pi) * -0.5))))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log(Float64(Float64(1.0 / expm1(Float64(Float64(f * pi) * 0.5))) + Float64(-1.0 / expm1(Float64(Float64(f * pi) * -0.5))))) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(N[(1.0 / N[(Exp[N[(N[(f * Pi), $MachinePrecision] * 0.5), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(Exp[N[(N[(f * Pi), $MachinePrecision] * -0.5), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

3. Simplified 98.6%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around inf 5.2%

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

   1. expm1-define 5.2%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\color{blue}{\mathsf{expm1}\left(0.5 \cdot \left(f \cdot \pi\right)\right)}} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   2. *-commutative 5.2%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\left(f \cdot \pi\right) \cdot 0.5}\right)} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   3. expm1-define 98.8%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{1}{\color{blue}{\mathsf{expm1}\left(-0.5 \cdot \left(f \cdot \pi\right)\right)}}\right)}{\pi} \]

   4. *-commutative 98.8%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\left(f \cdot \pi\right) \cdot -0.5}\right)}\right)}{\pi} \]

7. Simplified 98.8%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot -0.5\right)}\right)}{\pi}} \]

8. Final simplification 98.8%

   \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot -0.5\right)}\right)}{\pi} \]
9. Add Preprocessing

Alternative 2: 96.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{2 \cdot \frac{-1}{\pi} - f \cdot \left(0.5 + f \cdot \left(\pi \cdot 0.041666666666666664\right)\right)}{f}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log
    (-
     (/ 1.0 (expm1 (* (* f PI) 0.5)))
     (/
      (- (* 2.0 (/ -1.0 PI)) (* f (+ 0.5 (* f (* PI 0.041666666666666664)))))
      f)))
   PI)))

C

double code(double f) {
	return -4.0 * (log(((1.0 / expm1(((f * ((double) M_PI)) * 0.5))) - (((2.0 * (-1.0 / ((double) M_PI))) - (f * (0.5 + (f * (((double) M_PI) * 0.041666666666666664))))) / f))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log(((1.0 / Math.expm1(((f * Math.PI) * 0.5))) - (((2.0 * (-1.0 / Math.PI)) - (f * (0.5 + (f * (Math.PI * 0.041666666666666664))))) / f))) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log(((1.0 / math.expm1(((f * math.pi) * 0.5))) - (((2.0 * (-1.0 / math.pi)) - (f * (0.5 + (f * (math.pi * 0.041666666666666664))))) / f))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log(Float64(Float64(1.0 / expm1(Float64(Float64(f * pi) * 0.5))) - Float64(Float64(Float64(2.0 * Float64(-1.0 / pi)) - Float64(f * Float64(0.5 + Float64(f * Float64(pi * 0.041666666666666664))))) / f))) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(N[(1.0 / N[(Exp[N[(N[(f * Pi), $MachinePrecision] * 0.5), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(2.0 * N[(-1.0 / Pi), $MachinePrecision]), $MachinePrecision] - N[(f * N[(0.5 + N[(f * N[(Pi * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

3. Simplified 98.6%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around inf 5.2%

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

   1. expm1-define 5.2%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\color{blue}{\mathsf{expm1}\left(0.5 \cdot \left(f \cdot \pi\right)\right)}} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   2. *-commutative 5.2%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\left(f \cdot \pi\right) \cdot 0.5}\right)} - \frac{1}{e^{-0.5 \cdot \left(f \cdot \pi\right)} - 1}\right)}{\pi} \]

   3. expm1-define 98.8%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{1}{\color{blue}{\mathsf{expm1}\left(-0.5 \cdot \left(f \cdot \pi\right)\right)}}\right)}{\pi} \]

   4. *-commutative 98.8%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{1}{\mathsf{expm1}\left(\color{blue}{\left(f \cdot \pi\right) \cdot -0.5}\right)}\right)}{\pi} \]

7. Simplified 98.8%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot -0.5\right)}\right)}{\pi}} \]

8. Taylor expanded in f around 0 96.8%

   \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \color{blue}{\frac{f \cdot \left(-1 \cdot \left(f \cdot \left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right)\right) - 0.5\right) - 2 \cdot \frac{1}{\pi}}{f}}\right)}{\pi} \]
9. Step-by-step derivation

   1. *-un-lft-identity 96.8%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{f \cdot \left(-1 \cdot \left(f \cdot \color{blue}{\left(1 \cdot \left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right)\right)}\right) - 0.5\right) - 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

   2. distribute-rgt-out 96.8%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{f \cdot \left(-1 \cdot \left(f \cdot \left(1 \cdot \color{blue}{\left(\pi \cdot \left(-0.08333333333333333 + 0.125\right)\right)}\right)\right) - 0.5\right) - 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

   3. metadata-eval 96.8%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{f \cdot \left(-1 \cdot \left(f \cdot \left(1 \cdot \left(\pi \cdot \color{blue}{0.041666666666666664}\right)\right)\right) - 0.5\right) - 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

10. Applied egg-rr 96.8%

    \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{f \cdot \left(-1 \cdot \left(f \cdot \color{blue}{\left(1 \cdot \left(\pi \cdot 0.041666666666666664\right)\right)}\right) - 0.5\right) - 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]
11. Step-by-step derivation

    1. *-lft-identity 96.8%

       \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{f \cdot \left(-1 \cdot \left(f \cdot \color{blue}{\left(\pi \cdot 0.041666666666666664\right)}\right) - 0.5\right) - 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

12. Simplified 96.8%

    \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{f \cdot \left(-1 \cdot \left(f \cdot \color{blue}{\left(\pi \cdot 0.041666666666666664\right)}\right) - 0.5\right) - 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

13. Final simplification 96.8%

    \[\leadsto -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{2 \cdot \frac{-1}{\pi} - f \cdot \left(0.5 + f \cdot \left(\pi \cdot 0.041666666666666664\right)\right)}{f}\right)}{\pi} \]
14. Add Preprocessing

Alternative 3: 96.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{\mathsf{fma}\left(f, f \cdot \left(\pi \cdot 0.08333333333333333\right), \frac{2}{\pi \cdot 0.5}\right)}{f}\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log (/ (fma f (* f (* PI 0.08333333333333333)) (/ 2.0 (* PI 0.5))) f))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log((fma(f, (f * (((double) M_PI) * 0.08333333333333333)), (2.0 / (((double) M_PI) * 0.5))) / f)) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(Float64(fma(f, Float64(f * Float64(pi * 0.08333333333333333)), Float64(2.0 / Float64(pi * 0.5))) / f)) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(f * N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $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 96.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right)\right)\right) + 2 \cdot \frac{1}{0.25 \cdot \pi - -0.25 \cdot \pi}}{f}\right)} \]

5. Simplified 96.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(f, f \cdot \mathsf{fma}\left(0.0625, \pi \cdot 2, \left(\left(2 \cdot \left(\pi \cdot 2\right)\right) \cdot 0.005208333333333333\right) \cdot -2\right), \frac{2}{\pi \cdot 0.5}\right)}{f}\right)} \]

6. Taylor expanded in f around 0 96.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f, \color{blue}{f \cdot \left(-0.041666666666666664 \cdot \pi + 0.125 \cdot \pi\right)}, \frac{2}{\pi \cdot 0.5}\right)}{f}\right) \]
7. Step-by-step derivation

   1. distribute-rgt-out 96.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f, f \cdot \color{blue}{\left(\pi \cdot \left(-0.041666666666666664 + 0.125\right)\right)}, \frac{2}{\pi \cdot 0.5}\right)}{f}\right) \]

   2. metadata-eval 96.6%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f, f \cdot \left(\pi \cdot \color{blue}{0.08333333333333333}\right), \frac{2}{\pi \cdot 0.5}\right)}{f}\right) \]

8. Simplified 96.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f, \color{blue}{f \cdot \left(\pi \cdot 0.08333333333333333\right)}, \frac{2}{\pi \cdot 0.5}\right)}{f}\right) \]

9. Final simplification 96.6%

   \[\leadsto \log \left(\frac{\mathsf{fma}\left(f, f \cdot \left(\pi \cdot 0.08333333333333333\right), \frac{2}{\pi \cdot 0.5}\right)}{f}\right) \cdot \frac{-1}{\frac{\pi}{4}} \]
10. Add Preprocessing

Alternative 4: 95.9% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

3. Simplified 98.6%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 95.9%

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

   1. mul-1-neg 95.9%

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

   2. unsub-neg 95.9%

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

7. Simplified 95.9%

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

   1. associate-*r/ 95.9%

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

   2. diff-log 96.0%

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

9. Applied egg-rr 96.0%

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

    1. associate-*r/ 95.8%

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

    2. *-commutative 95.8%

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

    3. associate-/l/ 95.8%

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

    4. associate-/r* 95.8%

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

11. Simplified 95.8%

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

12. Final simplification 95.8%

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

Alternative 5: 96.1% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

3. Simplified 98.6%

   \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi}} \]
4. Add Preprocessing

5. Taylor expanded in f around 0 95.9%

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

   1. mul-1-neg 95.9%

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

   2. unsub-neg 95.9%

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

7. Simplified 95.9%

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

   1. associate-*r/ 95.9%

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

   2. diff-log 96.0%

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

9. Applied egg-rr 96.0%

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

10. Final simplification 96.0%

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

Reproduce

herbie shell --seed 2024075 
(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)))))))))