VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.0%

Time: 15.9s

Alternatives: 6

Speedup: 4.9×

• could not determine a ground truth

  1. f = 1452281981.7059193

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: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Python

def code(f):
	return -4.0 * (math.log(((1.0 / math.expm1((0.5 * (math.pi * f)))) + (-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(0.5 * Float64(pi * f)))) + 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[(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] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

3. Simplified 99.0%

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

5. Taylor expanded in f around inf 4.4%

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

   1. expm1-define 4.6%

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

   2. *-commutative 4.6%

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

   3. expm1-define 99.1%

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

   4. associate-*r* 99.1%

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

   5. *-commutative 99.1%

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

   6. *-commutative 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 2: 96.4% 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 6.5%

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

3. Simplified 99.0%

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

5. Taylor expanded in f around inf 4.4%

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

   1. expm1-define 4.6%

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

   2. *-commutative 4.6%

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

   3. expm1-define 99.1%

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

   4. associate-*r* 99.1%

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

   5. *-commutative 99.1%

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

   6. *-commutative 99.1%

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

7. Simplified 99.1%

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

8. Taylor expanded in f around 0 98.1%

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

9. Final simplification 98.1%

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

Alternative 3: 96.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

3. Simplified 99.0%

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

5. Taylor expanded in f around 0 98.1%

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

   1. mul-1-neg 98.1%

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

   2. unsub-neg 98.1%

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

7. Simplified 98.1%

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

   1. expm1-log1p-u 96.9%

      \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\right)\right)} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \]

   2. expm1-undefine 96.9%

      \[\leadsto -4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\right)} - 1\right)} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \]

   3. diff-log 97.0%

      \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\log \left(\frac{\frac{4}{\pi}}{f}\right)}}{\pi}\right)} - 1\right) - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \]

9. Applied egg-rr 97.0%

   \[\leadsto -4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}\right)} - 1\right)} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \]
10. Step-by-step derivation

    1. expm1-define 97.0%

       \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}\right)\right)} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \]

11. Simplified 97.0%

    \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}\right)\right)} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \]
12. Step-by-step derivation

    1. expm1-undefine 97.0%

       \[\leadsto -4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}\right)} - 1\right)} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \]

    2. log1p-expm1-u 96.9%

       \[\leadsto -4 \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}\right)\right)\right)}} - 1\right) - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \]

    3. log1p-undefine 97.0%

       \[\leadsto -4 \cdot \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}\right)\right)\right)}} - 1\right) - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \]

    4. rem-exp-log 97.0%

       \[\leadsto -4 \cdot \left(\color{blue}{\left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}\right)\right)\right)} - 1\right) - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \]

    5. expm1-log1p-u 98.1%

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

    6. associate-/l/ 98.1%

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

13. Applied egg-rr 98.1%

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

14. Final simplification 98.1%

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

Alternative 4: 96.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

3. Simplified 99.0%

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

5. Taylor expanded in f around 0 98.1%

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

   1. mul-1-neg 98.1%

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

   2. unsub-neg 98.1%

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

7. Simplified 98.1%

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

   1. associate-*r/ 98.1%

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

   2. diff-log 98.1%

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

9. Applied egg-rr 98.1%

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

    1. associate-/l/ 98.1%

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

    2. *-commutative 98.1%

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

11. Simplified 98.1%

    \[\leadsto \color{blue}{\frac{-4 \cdot \log \left(\frac{4}{\pi \cdot f}\right)}{\pi}} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \]
12. Add Preprocessing

Alternative 5: 95.8% 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 6.5%

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

3. Simplified 99.0%

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

5. Taylor expanded in f around inf 4.4%

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

   1. expm1-define 4.6%

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

   2. *-commutative 4.6%

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

   3. expm1-define 99.1%

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

   4. associate-*r* 99.1%

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

   5. *-commutative 99.1%

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

   6. *-commutative 99.1%

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

7. Simplified 99.1%

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

8. Taylor expanded in f around 0 97.6%

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

   1. associate-/l/ 97.6%

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

10. Simplified 97.6%

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

Alternative 6: 4.2% accurate, 76.0× speedup

Math

\[\begin{array}{l} \\ -0.08333333333333333 \cdot \left(\pi \cdot \left(f \cdot f\right)\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* -0.08333333333333333 (* PI (* f f))))

C

double code(double f) {
	return -0.08333333333333333 * (((double) M_PI) * (f * f));
}

Java

public static double code(double f) {
	return -0.08333333333333333 * (Math.PI * (f * f));
}

Python

def code(f):
	return -0.08333333333333333 * (math.pi * (f * f))

Julia

function code(f)
	return Float64(-0.08333333333333333 * Float64(pi * Float64(f * f)))
end

MATLAB

function tmp = code(f)
	tmp = -0.08333333333333333 * (pi * (f * f));
end

Wolfram

code[f_] := N[(-0.08333333333333333 * N[(Pi * N[(f * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

3. Simplified 99.0%

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

5. Taylor expanded in f around 0 98.1%

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

   1. mul-1-neg 98.1%

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

   2. unsub-neg 98.1%

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

7. Simplified 98.1%

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

   1. expm1-log1p-u 96.9%

      \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\right)\right)} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \]

   2. expm1-undefine 96.9%

      \[\leadsto -4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}\right)} - 1\right)} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \]

   3. diff-log 97.0%

      \[\leadsto -4 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\log \left(\frac{\frac{4}{\pi}}{f}\right)}}{\pi}\right)} - 1\right) - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \]

9. Applied egg-rr 97.0%

   \[\leadsto -4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}\right)} - 1\right)} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \]
10. Step-by-step derivation

    1. expm1-define 97.0%

       \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}\right)\right)} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \]

11. Simplified 97.0%

    \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{\frac{4}{\pi}}{f}\right)}{\pi}\right)\right)} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \]

12. Taylor expanded in f around inf 4.2%

    \[\leadsto \color{blue}{-0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)} \]
13. Step-by-step derivation

    1. unpow2 4.2%

       \[\leadsto -0.08333333333333333 \cdot \left(\color{blue}{\left(f \cdot f\right)} \cdot \pi\right) \]

14. Applied egg-rr 4.2%

    \[\leadsto -0.08333333333333333 \cdot \left(\color{blue}{\left(f \cdot f\right)} \cdot \pi\right) \]

15. Final simplification 4.2%

    \[\leadsto -0.08333333333333333 \cdot \left(\pi \cdot \left(f \cdot f\right)\right) \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024182 
(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)))))))))