VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 98.9%

Time: 24.4s

Alternatives: 7

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: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-4 \cdot \log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(f \cdot \sqrt[3]{{\pi}^{3}}\right) \cdot -0.5\right)}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (*
   -4.0
   (log
    (+
     (/ 1.0 (expm1 (* (* f PI) 0.5)))
     (/ -1.0 (expm1 (* (* f (cbrt (pow PI 3.0))) -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 * cbrt(pow(((double) M_PI), 3.0))) * -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.cbrt(Math.pow(Math.PI, 3.0))) * -0.5)))))) / Math.PI;
}

Julia

function code(f)
	return Float64(Float64(-4.0 * log(Float64(Float64(1.0 / expm1(Float64(Float64(f * pi) * 0.5))) + Float64(-1.0 / expm1(Float64(Float64(f * cbrt((pi ^ 3.0))) * -0.5)))))) / pi)
end

Wolfram

code[f_] := N[(N[(-4.0 * 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 * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

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 4.2%

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

   1. associate-*r/ 4.2%

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

   2. expm1-define 4.4%

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

   3. *-commutative 4.4%

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

   4. expm1-define 98.7%

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

   5. *-commutative 98.7%

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

7. Simplified 98.7%

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

   1. add-cbrt-cube 98.7%

      \[\leadsto \frac{-4 \cdot \log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{1}{\mathsf{expm1}\left(\left(f \cdot \color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}\right) \cdot -0.5\right)}\right)}{\pi} \]

   2. pow3 98.7%

      \[\leadsto \frac{-4 \cdot \log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{1}{\mathsf{expm1}\left(\left(f \cdot \sqrt[3]{\color{blue}{{\pi}^{3}}}\right) \cdot -0.5\right)}\right)}{\pi} \]

9. Applied egg-rr 98.7%

   \[\leadsto \frac{-4 \cdot \log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} - \frac{1}{\mathsf{expm1}\left(\left(f \cdot \color{blue}{\sqrt[3]{{\pi}^{3}}}\right) \cdot -0.5\right)}\right)}{\pi} \]

10. Final simplification 98.7%

    \[\leadsto \frac{-4 \cdot \log \left(\frac{1}{\mathsf{expm1}\left(\left(f \cdot \pi\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(f \cdot \sqrt[3]{{\pi}^{3}}\right) \cdot -0.5\right)}\right)}{\pi} \]
11. Add Preprocessing

Alternative 2: 98.9% accurate, 1.7× speedup

Math

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

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(Float64(-4.0 * 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[(N[(-4.0 * 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]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

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 4.2%

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

   1. associate-*r/ 4.2%

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

   2. expm1-define 4.4%

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

   3. *-commutative 4.4%

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

   4. expm1-define 98.7%

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

   5. *-commutative 98.7%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 3: 98.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

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. Final simplification 98.6%

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

Alternative 4: 96.2% accurate, 2.3× speedup

Math

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

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(Float64(-4.0 * 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[(N[(-4.0 * 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]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

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 4.2%

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

   1. associate-*r/ 4.2%

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

   2. expm1-define 4.4%

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

   3. *-commutative 4.4%

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

   4. expm1-define 98.7%

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

   5. *-commutative 98.7%

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

7. Simplified 98.7%

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

8. Taylor expanded in f around 0 97.2%

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

   1. distribute-rgt-out 97.2%

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

   2. metadata-eval 97.2%

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

10. Applied egg-rr 97.2%

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

11. Final simplification 97.2%

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

Alternative 5: 96.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(f)
	return Float64(Float64(-4.0 / pi) * 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(Float64(f * pi) * 0.041666666666666664)))) / f))))
end

Wolfram

code[f_] := N[(N[(-4.0 / Pi), $MachinePrecision] * 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[(N[(f * Pi), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

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 97.0%

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

   1. distribute-rgt-out 97.0%

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

   2. metadata-eval 97.0%

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

7. Applied egg-rr 97.0%

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

8. Taylor expanded in f around 0 97.0%

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

9. Final simplification 97.0%

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

Alternative 6: 95.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

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 4.2%

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

   1. associate-*r/ 4.2%

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

   2. expm1-define 4.4%

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

   3. *-commutative 4.4%

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

   4. expm1-define 98.7%

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

   5. *-commutative 98.7%

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

7. Simplified 98.7%

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

8. Taylor expanded in f around 0 97.2%

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

9. Taylor expanded in f around 0 96.8%

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

    1. *-commutative 96.8%

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

    2. associate-*r/ 96.8%

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

    3. metadata-eval 96.8%

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

11. Simplified 96.8%

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

12. Final simplification 96.8%

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

Alternative 7: 95.7% accurate, 4.9× speedup

Math

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

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(-4.0 * Float64(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[(-4.0 * N[(N[Log[N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

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 96.7%

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

   1. mul-1-neg 96.7%

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

   2. unsub-neg 96.7%

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

7. Simplified 96.7%

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

   1. *-un-lft-identity 96.7%

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

   2. diff-log 96.8%

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

9. Applied egg-rr 96.8%

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

    1. *-lft-identity 96.8%

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

11. Simplified 96.8%

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

Reproduce

herbie shell --seed 2024107 
(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)))))))))