VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 99.0%

Time: 18.0s

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: 6.8% 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

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

FPCore

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

C

double code(double f) {
	return (-4.0 * log(((1.0 / expm1((((double) M_PI) * (f * 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((Math.PI * (f * 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((math.pi * (f * 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(pi * Float64(f * 0.5)))) + Float64(-1.0 / expm1(Float64(f * Float64(pi * -0.5))))))) / pi)
end

Wolfram

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

Derivation

1. Initial program 8.6%

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

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 6.6%

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

   1. associate-*r/ 6.6%

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

7. Simplified 98.8%

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

Alternative 2: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Python

def code(f):
	return math.log(((1.0 / math.expm1((0.5 * (math.pi * f)))) + (-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(0.5 * Float64(pi * f)))) + 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[(0.5 * N[(Pi * f), $MachinePrecision]), $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 8.6%

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

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%

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

Alternative 3: 96.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

double code(double f) {
	return (-4.0 * log((((pow(f, 2.0) * (((((double) M_PI) * -0.08333333333333333) + (((double) M_PI) * 0.125)) - ((((double) M_PI) * -0.125) + (((double) M_PI) * 0.08333333333333333)))) + (4.0 * (1.0 / ((double) M_PI)))) / f))) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (-4.0 * Math.log((((Math.pow(f, 2.0) * (((Math.PI * -0.08333333333333333) + (Math.PI * 0.125)) - ((Math.PI * -0.125) + (Math.PI * 0.08333333333333333)))) + (4.0 * (1.0 / Math.PI))) / f))) / Math.PI;
}

Python

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

Julia

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

MATLAB

function tmp = code(f)
	tmp = (-4.0 * log(((((f ^ 2.0) * (((pi * -0.08333333333333333) + (pi * 0.125)) - ((pi * -0.125) + (pi * 0.08333333333333333)))) + (4.0 * (1.0 / pi))) / f))) / pi;
end

Wolfram

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

Derivation

1. Initial program 8.6%

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

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 6.6%

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

   1. associate-*r/ 6.6%

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

7. Simplified 98.8%

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

8. Taylor expanded in f around 0 96.5%

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

9. Final simplification 96.5%

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

Alternative 4: 96.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.6%

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

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 96.4%

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

   1. pow1 96.4%

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

   2. mul-1-neg 96.4%

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

   3. distribute-rgt-out 96.4%

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

   4. metadata-eval 96.4%

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

7. Applied egg-rr 96.4%

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

   1. unpow1 96.4%

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

   2. distribute-rgt-neg-in 96.4%

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

   3. distribute-rgt-neg-in 96.4%

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

   4. metadata-eval 96.4%

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

9. Simplified 96.4%

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

10. Final simplification 96.4%

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

Alternative 5: 95.7% accurate, 4.9× speedup

Math

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

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

Derivation

1. Initial program 8.6%

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

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 6.6%

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

   1. associate-*r/ 6.6%

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

7. Simplified 98.8%

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

8. Taylor expanded in f around 0 95.9%

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

   1. *-commutative 95.9%

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

10. Simplified 95.9%

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

Alternative 6: 95.5% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.6%

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

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 95.8%

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

   1. *-commutative 95.8%

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

7. Simplified 95.8%

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

8. Final simplification 95.8%

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

Alternative 7: 3.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \log 0 \end{array} \]

FPCore

(FPCore (f) :precision binary64 (log 0.0))

C

double code(double f) {
	return log(0.0);
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = log(0.0d0)
end function

Java

public static double code(double f) {
	return Math.log(0.0);
}

Python

def code(f):
	return math.log(0.0)

Julia

function code(f)
	return log(0.0)
end

MATLAB

function tmp = code(f)
	tmp = log(0.0);
end

Wolfram

code[f_] := N[Log[0.0], $MachinePrecision]

Derivation

1. Initial program 8.6%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 8.6%

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

   2. distribute-neg-frac2 8.6%

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

   3. associate-*l/ 8.6%

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

3. Simplified 8.6%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{{\left(e^{\frac{\pi}{-4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{f}}{{\left(e^{\frac{\pi}{4}}\right)}^{f} - {\left(e^{\frac{\pi}{-4}}\right)}^{f}}\right)}{\frac{\pi}{-4}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 8.6%

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

   2. log-prod 8.6%

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

   3. metadata-eval 8.6%

      \[\leadsto \frac{\color{blue}{0} + \log \left(\frac{{\left(e^{\frac{\pi}{-4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{f}}{{\left(e^{\frac{\pi}{4}}\right)}^{f} - {\left(e^{\frac{\pi}{-4}}\right)}^{f}}\right)}{\frac{\pi}{-4}} \]

6. Applied egg-rr 3.1%

   \[\leadsto \frac{\color{blue}{0 + \log \left(\frac{{\left({\left(e^{\pi}\right)}^{-0.25}\right)}^{f} + {\left({\left(e^{\pi}\right)}^{-0.25}\right)}^{f}}{{\left({\left(e^{\pi}\right)}^{-0.25}\right)}^{f} - {\left({\left(e^{\pi}\right)}^{-0.25}\right)}^{f}}\right)}}{\frac{\pi}{-4}} \]

8. Simplified 0.7%

   \[\leadsto \frac{\color{blue}{\log 0}}{\frac{\pi}{-4}} \]
9. Step-by-step derivation

   1. add-log-exp 0.7%

      \[\leadsto \color{blue}{\log \left(e^{\frac{\log 0}{\frac{\pi}{-4}}}\right)} \]

   2. div-inv 0.7%

      \[\leadsto \log \left(e^{\color{blue}{\log 0 \cdot \frac{1}{\frac{\pi}{-4}}}}\right) \]

   3. clear-num 0.7%

      \[\leadsto \log \left(e^{\log 0 \cdot \color{blue}{\frac{-4}{\pi}}}\right) \]

   4. exp-to-pow 0.7%

      \[\leadsto \log \color{blue}{\left({0}^{\left(\frac{-4}{\pi}\right)}\right)} \]

10. Applied egg-rr 0.7%

    \[\leadsto \color{blue}{\log \left({0}^{\left(\frac{-4}{\pi}\right)}\right)} \]
11. Step-by-step derivation

    1. pow-base-0 3.1%

       \[\leadsto \log \color{blue}{0} \]

12. Simplified 3.1%

    \[\leadsto \color{blue}{\log 0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024146 
(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)))))))))