VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.0%

Time: 31.0s

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: 6.7% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around inf 5.8%

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

   1. associate-*r/ 5.8%

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

   2. expm1-define 6.0%

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

   3. *-commutative 6.0%

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

   4. expm1-define 99.1%

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

   5. *-commutative 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 2: 98.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double f) {
	return log(((1.0 / expm1((0.5 * (f * ((double) M_PI))))) + (-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 * (f * Math.PI)))) + (-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 * (f * math.pi)))) + (-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(f * pi)))) + 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[(f * Pi), $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 6.7%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 3: 96.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double f) {
	return (-4.0 * ((log((4.0 / ((double) M_PI))) - log(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)) - Math.log(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)) - math.log(f)) / math.pi)) - (math.pow(f, 2.0) * (math.pi * 0.08333333333333333))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around 0 96.7%

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

   1. mul-1-neg 96.7%

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

   2. unsub-neg 96.7%

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

   3. mul-1-neg 96.7%

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

   4. unsub-neg 96.7%

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

   5. distribute-rgt-out 96.7%

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

   6. distribute-rgt-out 96.7%

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

   7. metadata-eval 96.7%

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

7. Simplified 96.7%

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

8. Final simplification 96.7%

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

Alternative 4: 96.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around 0 96.7%

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

   1. mul-1-neg 96.7%

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

   2. unsub-neg 96.7%

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

   3. mul-1-neg 96.7%

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

   4. unsub-neg 96.7%

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

   5. distribute-rgt-out 96.7%

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

   6. distribute-rgt-out 96.7%

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

   7. metadata-eval 96.7%

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

7. Simplified 96.7%

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

   1. *-un-lft-identity 96.7%

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

   2. diff-log 96.7%

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

9. Applied egg-rr 96.7%

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

    1. *-lft-identity 96.7%

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

    2. associate-/r* 96.7%

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

11. Simplified 96.7%

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

12. Final simplification 96.7%

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

Alternative 5: 96.2% 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 6.7%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around 0 96.3%

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

   1. mul-1-neg 96.3%

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

   2. unsub-neg 96.3%

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

7. Simplified 96.3%

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

8. Final simplification 96.3%

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

Alternative 6: 96.1% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

3. Taylor expanded in f around 0 96.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{2}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right)} \]
4. Step-by-step derivation

   1. distribute-rgt-out-- 96.1%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{2}{f \cdot \color{blue}{\left(\pi \cdot \left(0.25 - -0.25\right)\right)}}\right) \]

   2. metadata-eval 96.1%

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

5. Simplified 96.1%

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

   1. associate-*l/ 96.3%

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

   2. *-un-lft-identity 96.3%

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

   3. associate-*r* 96.3%

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

   4. *-commutative 96.3%

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

   5. associate-*l* 96.3%

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

   6. div-inv 96.3%

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

   7. metadata-eval 96.3%

      \[\leadsto -\frac{\log \left(\frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)}{\pi \cdot \color{blue}{0.25}} \]

7. Applied egg-rr 96.3%

   \[\leadsto -\color{blue}{\frac{\log \left(\frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)}{\pi \cdot 0.25}} \]
8. Step-by-step derivation

   1. *-lft-identity 96.3%

      \[\leadsto -\frac{\color{blue}{1 \cdot \log \left(\frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)}}{\pi \cdot 0.25} \]

   2. *-commutative 96.3%

      \[\leadsto -\frac{1 \cdot \log \left(\frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)}{\color{blue}{0.25 \cdot \pi}} \]

   3. times-frac 96.3%

      \[\leadsto -\color{blue}{\frac{1}{0.25} \cdot \frac{\log \left(\frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)}{\pi}} \]

   4. metadata-eval 96.3%

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

   5. associate-*r* 96.3%

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

   6. *-commutative 96.3%

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

   7. *-commutative 96.3%

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

   8. associate-/r* 96.3%

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

   9. metadata-eval 96.3%

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

   10. *-commutative 96.3%

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

9. Simplified 96.3%

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

    1. associate-/r* 96.3%

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

    2. div-inv 96.3%

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

11. Applied egg-rr 96.3%

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

12. Taylor expanded in f around 0 96.3%

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

    1. associate-/r* 96.3%

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

14. Simplified 96.3%

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

15. Final simplification 96.3%

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

Alternative 7: 96.1% 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 6.7%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around 0 96.3%

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

   1. mul-1-neg 96.3%

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

   2. unsub-neg 96.3%

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

7. Simplified 96.3%

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

   1. *-un-lft-identity 96.3%

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

   2. *-un-lft-identity 96.3%

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

   3. prod-diff 96.3%

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

   4. *-commutative 96.3%

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

   5. *-un-lft-identity 96.3%

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

   6. fma-neg 96.3%

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

   7. *-un-lft-identity 96.3%

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

   8. diff-log 96.3%

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

   9. associate-/r* 96.3%

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

   10. *-commutative 96.3%

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

   11. *-un-lft-identity 96.3%

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

9. Applied egg-rr 96.3%

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

    1. fma-undefine 96.3%

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

    2. *-rgt-identity 96.3%

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

    3. +-commutative 96.3%

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

    4. sub-neg 96.3%

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

    5. +-inverses 96.3%

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

    6. +-rgt-identity 96.3%

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

    7. associate-/r* 96.3%

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

11. Simplified 96.3%

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

12. Final simplification 96.3%

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

Reproduce

herbie shell --seed 2024072 
(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)))))))))