VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.1% → 99.0%

Time: 28.9s

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: 7.1% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around inf 6.2%

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

7. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. add-sqr-sqrt 99.3%

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

   4. pow2 99.3%

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

   5. *-commutative 99.3%

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

   6. *-commutative 99.3%

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

9. Applied egg-rr 99.3%

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

10. Final simplification 99.3%

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

Alternative 2: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around inf 6.2%

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

7. Simplified 99.3%

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

8. Final simplification 99.3%

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

Alternative 3: 96.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{1}{\mathsf{expm1}\left(f \cdot \left(\pi \cdot 0.5\right)\right)} + \frac{f \cdot \left(0.5 - f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log
    (+
     (/ 1.0 (expm1 (* f (* PI 0.5))))
     (/
      (+
       (* f (- 0.5 (* f (+ (* PI -0.125) (* PI 0.08333333333333333)))))
       (* 2.0 (/ 1.0 PI)))
      f)))
   PI)))

C

double code(double f) {
	return -4.0 * (log(((1.0 / expm1((f * (((double) M_PI) * 0.5)))) + (((f * (0.5 - (f * ((((double) M_PI) * -0.125) + (((double) M_PI) * 0.08333333333333333))))) + (2.0 * (1.0 / ((double) M_PI)))) / 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)))) + (((f * (0.5 - (f * ((Math.PI * -0.125) + (Math.PI * 0.08333333333333333))))) + (2.0 * (1.0 / Math.PI))) / f))) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log(((1.0 / math.expm1((f * (math.pi * 0.5)))) + (((f * (0.5 - (f * ((math.pi * -0.125) + (math.pi * 0.08333333333333333))))) + (2.0 * (1.0 / math.pi))) / f))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log(Float64(Float64(1.0 / expm1(Float64(f * Float64(pi * 0.5)))) + Float64(Float64(Float64(f * Float64(0.5 - Float64(f * Float64(Float64(pi * -0.125) + Float64(pi * 0.08333333333333333))))) + Float64(2.0 * Float64(1.0 / pi))) / f))) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(N[(1.0 / N[(Exp[N[(f * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(f * N[(0.5 - N[(f * N[(N[(Pi * -0.125), $MachinePrecision] + N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around inf 6.2%

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

7. Simplified 99.3%

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

8. Taylor expanded in f around 0 96.8%

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

9. Final simplification 96.8%

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

Alternative 4: 95.7% accurate, 2.4× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around inf 6.2%

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

7. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. add-sqr-sqrt 99.3%

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

   4. pow2 99.3%

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

   5. *-commutative 99.3%

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

   6. *-commutative 99.3%

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

9. Applied egg-rr 99.3%

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

10. Taylor expanded in f around 0 0.0%

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

    1. unpow2 0.0%

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

    2. rem-square-sqrt 0.0%

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

    3. associate-*r/ 0.0%

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

    4. metadata-eval 0.0%

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

    5. *-commutative 0.0%

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

    6. unpow2 0.0%

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

    7. rem-square-sqrt 96.1%

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

    8. associate-*l* 96.1%

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

    9. metadata-eval 96.1%

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

12. Simplified 96.1%

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

13. Final simplification 96.1%

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

Alternative 5: 95.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around 0 96.1%

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

   1. mul-1-neg 96.1%

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

   2. unsub-neg 96.1%

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

7. Simplified 96.1%

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

8. Final simplification 96.1%

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

Alternative 6: 95.7% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around 0 96.1%

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

   1. mul-1-neg 96.1%

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

   2. unsub-neg 96.1%

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

7. Simplified 96.1%

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

   1. *-un-lft-identity 96.1%

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

   2. diff-log 96.1%

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

9. Applied egg-rr 96.1%

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

    1. *-lft-identity 96.1%

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

    2. associate-/r* 96.1%

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

11. Simplified 96.1%

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

12. Final simplification 96.1%

    \[\leadsto -4 \cdot \frac{\log \left(\frac{4}{f \cdot \pi}\right)}{\pi} \]
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 7.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around 0 96.1%

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

   1. mul-1-neg 96.1%

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

   2. unsub-neg 96.1%

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

7. Simplified 96.1%

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

   1. diff-log 96.1%

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

9. Applied egg-rr 96.1%

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

10. Final simplification 96.1%

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

Reproduce

herbie shell --seed 2024054 
(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)))))))))