VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.1% → 99.1%

Time: 18.8s

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: 99.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

3. Simplified 99.3%

   \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
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. expm1-define 6.6%

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

   2. associate-*r* 6.6%

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

   3. *-commutative 6.6%

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

   4. *-commutative 6.6%

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

   5. expm1-define 99.4%

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

   6. *-commutative 99.4%

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

   7. *-commutative 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 2: 96.6% accurate, 2.3× speedup

Math

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

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

Derivation

1. Initial program 7.7%

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

3. Simplified 99.3%

   \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
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. expm1-define 6.6%

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

   2. associate-*r* 6.6%

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

   3. *-commutative 6.6%

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

   4. *-commutative 6.6%

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

   5. expm1-define 99.4%

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

   6. *-commutative 99.4%

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

   7. *-commutative 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in f around 0 97.5%

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

9. Final simplification 97.5%

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

Alternative 3: 96.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

3. Simplified 99.3%

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

5. Taylor expanded in f around 0 97.4%

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

   1. pow1 97.4%

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

   2. mul-1-neg 97.4%

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

   3. distribute-rgt-out 97.4%

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

   4. metadata-eval 97.4%

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

7. Applied egg-rr 97.4%

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

   1. unpow1 97.4%

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

   2. distribute-rgt-neg-in 97.4%

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

   3. distribute-rgt-neg-in 97.4%

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

   4. metadata-eval 97.4%

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

9. Simplified 97.4%

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

10. Final simplification 97.4%

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

Alternative 4: 96.4% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

3. Simplified 99.3%

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

5. Taylor expanded in f around 0 97.4%

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

   1. pow1 97.4%

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

   2. distribute-rgt-out 97.4%

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

   3. fmm-def 97.4%

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

   4. metadata-eval 97.4%

      \[\leadsto \log \left(\frac{{\left({f}^{2} \cdot \mathsf{fma}\left(\pi, \color{blue}{0.041666666666666664}, -\left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)}^{1} + 4 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   5. distribute-rgt-out 97.4%

      \[\leadsto \log \left(\frac{{\left({f}^{2} \cdot \mathsf{fma}\left(\pi, 0.041666666666666664, -\color{blue}{\pi \cdot \left(-0.125 + 0.08333333333333333\right)}\right)\right)}^{1} + 4 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

   6. metadata-eval 97.4%

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

7. Applied egg-rr 97.4%

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

   1. unpow1 97.4%

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

   2. fmm-undef 97.4%

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

   3. distribute-lft-out-- 97.4%

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

   4. metadata-eval 97.4%

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

9. Simplified 97.4%

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

10. Final simplification 97.4%

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

Alternative 5: 96.1% 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 7.7%

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

3. Simplified 99.3%

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

5. Taylor expanded in f around 0 97.0%

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

   1. mul-1-neg 97.0%

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

   2. unsub-neg 97.0%

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

7. Simplified 97.0%

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

Alternative 6: 95.9% accurate, 4.9× speedup

Math

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

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

Derivation

1. Initial program 7.7%

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

3. Simplified 99.3%

   \[\leadsto \color{blue}{\log \left(\frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot \pi\right) \cdot f\right)} + \frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot \pi\right) \cdot f\right)}\right) \cdot \frac{-4}{\pi}} \]
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. expm1-define 6.6%

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

   2. associate-*r* 6.6%

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

   3. *-commutative 6.6%

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

   4. *-commutative 6.6%

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

   5. expm1-define 99.4%

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

   6. *-commutative 99.4%

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

   7. *-commutative 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in f around 0 97.0%

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

   1. associate-/r* 97.0%

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

10. Simplified 97.0%

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

Alternative 7: 0.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \frac{-4}{\pi} \cdot \log 0 \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ -4.0 PI) (log 0.0)))

C

double code(double f) {
	return (-4.0 / ((double) M_PI)) * log(0.0);
}

Java

public static double code(double f) {
	return (-4.0 / Math.PI) * Math.log(0.0);
}

Python

def code(f):
	return (-4.0 / math.pi) * math.log(0.0)

Julia

function code(f)
	return Float64(Float64(-4.0 / pi) * log(0.0))
end

MATLAB

function tmp = code(f)
	tmp = (-4.0 / pi) * log(0.0);
end

Wolfram

code[f_] := N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[0.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

3. Simplified 99.3%

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

6. Applied egg-rr 0.7%

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

   1. +-inverses 0.7%

      \[\leadsto \log \color{blue}{0} \cdot \frac{-4}{\pi} \]

8. Simplified 0.7%

   \[\leadsto \log \color{blue}{0} \cdot \frac{-4}{\pi} \]

9. Final simplification 0.7%

   \[\leadsto \frac{-4}{\pi} \cdot \log 0 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(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)))))))))