VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.5% → 98.8%

Time: 26.5s

Alternatives: 6

Speedup: 2.5×

• 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: 6.5% 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: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot f\right) \cdot \pi\right)} + \left(-1 + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\right) \cdot \frac{-4}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log1p
   (+
    (/ 1.0 (expm1 (* (* 0.5 f) PI)))
    (+ -1.0 (/ -1.0 (expm1 (* PI (* f -0.5)))))))
  (/ -4.0 PI)))

C

double code(double f) {
	return log1p(((1.0 / expm1(((0.5 * f) * ((double) M_PI)))) + (-1.0 + (-1.0 / expm1((((double) M_PI) * (f * -0.5))))))) * (-4.0 / ((double) M_PI));
}

Java

public static double code(double f) {
	return Math.log1p(((1.0 / Math.expm1(((0.5 * f) * Math.PI))) + (-1.0 + (-1.0 / Math.expm1((Math.PI * (f * -0.5))))))) * (-4.0 / Math.PI);
}

Python

def code(f):
	return math.log1p(((1.0 / math.expm1(((0.5 * f) * math.pi))) + (-1.0 + (-1.0 / math.expm1((math.pi * (f * -0.5))))))) * (-4.0 / math.pi)

Julia

function code(f)
	return Float64(log1p(Float64(Float64(1.0 / expm1(Float64(Float64(0.5 * f) * pi))) + Float64(-1.0 + Float64(-1.0 / expm1(Float64(pi * Float64(f * -0.5))))))) * Float64(-4.0 / pi))
end

Wolfram

code[f_] := N[(N[Log[1 + N[(N[(1.0 / N[(Exp[N[(N[(0.5 * f), $MachinePrecision] * Pi), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[(-1.0 / N[(Exp[N[(Pi * N[(f * -0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $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 98.5%

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

   1. add-cbrt-cube 34.3%

      \[\leadsto \log \left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\sqrt[3]{\left(\left(\pi \cdot f\right) \cdot \left(\pi \cdot f\right)\right) \cdot \left(\pi \cdot f\right)}} \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi} \]

   2. pow1/3 34.3%

      \[\leadsto \log \left(\frac{1}{\mathsf{expm1}\left(\color{blue}{{\left(\left(\left(\pi \cdot f\right) \cdot \left(\pi \cdot f\right)\right) \cdot \left(\pi \cdot f\right)\right)}^{0.3333333333333333}} \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi} \]

   3. pow3 34.3%

      \[\leadsto \log \left(\frac{1}{\mathsf{expm1}\left({\color{blue}{\left({\left(\pi \cdot f\right)}^{3}\right)}}^{0.3333333333333333} \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi} \]

6. Applied egg-rr 34.3%

   \[\leadsto \log \left(\frac{1}{\mathsf{expm1}\left(\color{blue}{{\left({\left(\pi \cdot f\right)}^{3}\right)}^{0.3333333333333333}} \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) \cdot \frac{-4}{\pi} \]
7. Step-by-step derivation

   1. log1p-expm1-u 34.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{1}{\mathsf{expm1}\left({\left({\left(\pi \cdot f\right)}^{3}\right)}^{0.3333333333333333} \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right)\right)\right)} \cdot \frac{-4}{\pi} \]

   2. expm1-undefine 34.3%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{1}{\mathsf{expm1}\left({\left({\left(\pi \cdot f\right)}^{3}\right)}^{0.3333333333333333} \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right)} - 1}\right) \cdot \frac{-4}{\pi} \]

   3. add-exp-log 34.3%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\frac{1}{\mathsf{expm1}\left({\left({\left(\pi \cdot f\right)}^{3}\right)}^{0.3333333333333333} \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right)} - 1\right) \cdot \frac{-4}{\pi} \]

   4. pow-pow 98.5%

      \[\leadsto \mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(\color{blue}{{\left(\pi \cdot f\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) - 1\right) \cdot \frac{-4}{\pi} \]

   5. metadata-eval 98.5%

      \[\leadsto \mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left({\left(\pi \cdot f\right)}^{\color{blue}{1}} \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) - 1\right) \cdot \frac{-4}{\pi} \]

   6. pow1 98.5%

      \[\leadsto \mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\left(\pi \cdot f\right)} \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\left(-0.5 \cdot f\right) \cdot \pi\right)}\right) - 1\right) \cdot \frac{-4}{\pi} \]

   7. *-commutative 98.5%

      \[\leadsto \mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\color{blue}{\pi \cdot \left(-0.5 \cdot f\right)}\right)}\right) - 1\right) \cdot \frac{-4}{\pi} \]

   8. *-commutative 98.5%

      \[\leadsto \mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \color{blue}{\left(f \cdot -0.5\right)}\right)}\right) - 1\right) \cdot \frac{-4}{\pi} \]

8. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right) - 1\right)} \cdot \frac{-4}{\pi} \]
9. Step-by-step derivation

   1. associate--l+ 98.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{1}{\mathsf{expm1}\left(\left(\pi \cdot f\right) \cdot 0.5\right)} + \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} - 1\right)}\right) \cdot \frac{-4}{\pi} \]

   2. *-commutative 98.7%

      \[\leadsto \mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(\color{blue}{0.5 \cdot \left(\pi \cdot f\right)}\right)} + \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} - 1\right)\right) \cdot \frac{-4}{\pi} \]

   3. *-commutative 98.7%

      \[\leadsto \mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \color{blue}{\left(f \cdot \pi\right)}\right)} + \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} - 1\right)\right) \cdot \frac{-4}{\pi} \]

   4. associate-*r* 98.7%

      \[\leadsto \mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(\color{blue}{\left(0.5 \cdot f\right) \cdot \pi}\right)} + \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} - 1\right)\right) \cdot \frac{-4}{\pi} \]

10. Simplified 98.7%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot f\right) \cdot \pi\right)} + \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} - 1\right)\right)} \cdot \frac{-4}{\pi} \]

11. Final simplification 98.7%

    \[\leadsto \mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(\left(0.5 \cdot f\right) \cdot \pi\right)} + \left(-1 + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\right) \cdot \frac{-4}{\pi} \]
12. Add Preprocessing

Alternative 2: 98.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[f_] := N[(N[(-4.0 / Pi), $MachinePrecision] * 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]), $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 98.5%

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

5. Final simplification 98.5%

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

Alternative 3: 96.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(f)
	tmp = (-4.0 * ((log((4.0 / pi)) - log(f)) / pi)) + (-0.08333333333333333 * (pi * (f ^ 2.0)));
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[(-0.08333333333333333 * N[(Pi * N[Power[f, 2.0], $MachinePrecision]), $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 98.5%

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

5. Taylor expanded in f around 0 97.2%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + 0.25 \cdot \left({f}^{2} \cdot \left(\pi \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]
6. Step-by-step derivation

   1. +-commutative 97.2%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(0.25 \cdot \left({f}^{2} \cdot \left(\pi \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)\right) + -1 \cdot \log f\right)}\right) \cdot \frac{-4}{\pi} \]

   2. mul-1-neg 97.2%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \left(0.25 \cdot \left({f}^{2} \cdot \left(\pi \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)\right) + \color{blue}{\left(-\log f\right)}\right)\right) \cdot \frac{-4}{\pi} \]

7. Simplified 97.2%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(0.25 \cdot \left({f}^{2} \cdot \left(0.08333333333333333 \cdot {\pi}^{2}\right)\right) - \log f\right)\right)} \cdot \frac{-4}{\pi} \]

8. Taylor expanded in f around 0 97.3%

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

9. Final simplification 97.3%

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

Alternative 4: 96.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{-4}{\pi} \cdot \left({\left(f \cdot \pi\right)}^{2} \cdot 0.020833333333333332 - \left(\log \left(\frac{f}{4}\right) + \log \pi\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ -4.0 PI)
  (-
   (* (pow (* f PI) 2.0) 0.020833333333333332)
   (+ (log (/ f 4.0)) (log PI)))))

C

double code(double f) {
	return (-4.0 / ((double) M_PI)) * ((pow((f * ((double) M_PI)), 2.0) * 0.020833333333333332) - (log((f / 4.0)) + log(((double) M_PI))));
}

Java

public static double code(double f) {
	return (-4.0 / Math.PI) * ((Math.pow((f * Math.PI), 2.0) * 0.020833333333333332) - (Math.log((f / 4.0)) + Math.log(Math.PI)));
}

Python

def code(f):
	return (-4.0 / math.pi) * ((math.pow((f * math.pi), 2.0) * 0.020833333333333332) - (math.log((f / 4.0)) + math.log(math.pi)))

Julia

function code(f)
	return Float64(Float64(-4.0 / pi) * Float64(Float64((Float64(f * pi) ^ 2.0) * 0.020833333333333332) - Float64(log(Float64(f / 4.0)) + log(pi))))
end

MATLAB

function tmp = code(f)
	tmp = (-4.0 / pi) * ((((f * pi) ^ 2.0) * 0.020833333333333332) - (log((f / 4.0)) + log(pi)));
end

Wolfram

code[f_] := N[(N[(-4.0 / Pi), $MachinePrecision] * N[(N[(N[Power[N[(f * Pi), $MachinePrecision], 2.0], $MachinePrecision] * 0.020833333333333332), $MachinePrecision] - N[(N[Log[N[(f / 4.0), $MachinePrecision]], $MachinePrecision] + N[Log[Pi], $MachinePrecision]), $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 98.5%

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

5. Taylor expanded in f around 0 97.2%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + 0.25 \cdot \left({f}^{2} \cdot \left(\pi \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]
6. Step-by-step derivation

   1. +-commutative 97.2%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(0.25 \cdot \left({f}^{2} \cdot \left(\pi \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)\right) + -1 \cdot \log f\right)}\right) \cdot \frac{-4}{\pi} \]

   2. mul-1-neg 97.2%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \left(0.25 \cdot \left({f}^{2} \cdot \left(\pi \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)\right) + \color{blue}{\left(-\log f\right)}\right)\right) \cdot \frac{-4}{\pi} \]

7. Simplified 97.2%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(0.25 \cdot \left({f}^{2} \cdot \left(0.08333333333333333 \cdot {\pi}^{2}\right)\right) - \log f\right)\right)} \cdot \frac{-4}{\pi} \]
8. Step-by-step derivation

   1. +-commutative 97.2%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot \left({f}^{2} \cdot \left(0.08333333333333333 \cdot {\pi}^{2}\right)\right) - \log f\right) + \log \left(\frac{4}{\pi}\right)\right)} \cdot \frac{-4}{\pi} \]

   2. log-div 97.2%

      \[\leadsto \left(\left(0.25 \cdot \left({f}^{2} \cdot \left(0.08333333333333333 \cdot {\pi}^{2}\right)\right) - \log f\right) + \color{blue}{\left(\log 4 - \log \pi\right)}\right) \cdot \frac{-4}{\pi} \]

   3. associate-+r- 97.0%

      \[\leadsto \color{blue}{\left(\left(\left(0.25 \cdot \left({f}^{2} \cdot \left(0.08333333333333333 \cdot {\pi}^{2}\right)\right) - \log f\right) + \log 4\right) - \log \pi\right)} \cdot \frac{-4}{\pi} \]

9. Applied egg-rr 97.0%

   \[\leadsto \color{blue}{\left(\left(\left(0.25 \cdot {\left(f \cdot \left(\pi \cdot \sqrt{0.08333333333333333}\right)\right)}^{2} - \log f\right) + \log 4\right) - \log \pi\right)} \cdot \frac{-4}{\pi} \]
10. Step-by-step derivation

    1. sub-neg 97.0%

       \[\leadsto \color{blue}{\left(\left(\left(0.25 \cdot {\left(f \cdot \left(\pi \cdot \sqrt{0.08333333333333333}\right)\right)}^{2} - \log f\right) + \log 4\right) + \left(-\log \pi\right)\right)} \cdot \frac{-4}{\pi} \]

    2. associate-+l- 97.0%

       \[\leadsto \left(\color{blue}{\left(0.25 \cdot {\left(f \cdot \left(\pi \cdot \sqrt{0.08333333333333333}\right)\right)}^{2} - \left(\log f - \log 4\right)\right)} + \left(-\log \pi\right)\right) \cdot \frac{-4}{\pi} \]

    3. associate-*r* 97.0%

       \[\leadsto \left(\left(0.25 \cdot {\color{blue}{\left(\left(f \cdot \pi\right) \cdot \sqrt{0.08333333333333333}\right)}}^{2} - \left(\log f - \log 4\right)\right) + \left(-\log \pi\right)\right) \cdot \frac{-4}{\pi} \]

    4. *-commutative 97.0%

       \[\leadsto \left(\left(0.25 \cdot {\left(\color{blue}{\left(\pi \cdot f\right)} \cdot \sqrt{0.08333333333333333}\right)}^{2} - \left(\log f - \log 4\right)\right) + \left(-\log \pi\right)\right) \cdot \frac{-4}{\pi} \]

    5. unpow-prod-down 97.0%

       \[\leadsto \left(\left(0.25 \cdot \color{blue}{\left({\left(\pi \cdot f\right)}^{2} \cdot {\left(\sqrt{0.08333333333333333}\right)}^{2}\right)} - \left(\log f - \log 4\right)\right) + \left(-\log \pi\right)\right) \cdot \frac{-4}{\pi} \]

    6. pow2 97.0%

       \[\leadsto \left(\left(0.25 \cdot \left({\left(\pi \cdot f\right)}^{2} \cdot \color{blue}{\left(\sqrt{0.08333333333333333} \cdot \sqrt{0.08333333333333333}\right)}\right) - \left(\log f - \log 4\right)\right) + \left(-\log \pi\right)\right) \cdot \frac{-4}{\pi} \]

    7. pow1/2 97.0%

       \[\leadsto \left(\left(0.25 \cdot \left({\left(\pi \cdot f\right)}^{2} \cdot \left(\color{blue}{{0.08333333333333333}^{0.5}} \cdot \sqrt{0.08333333333333333}\right)\right) - \left(\log f - \log 4\right)\right) + \left(-\log \pi\right)\right) \cdot \frac{-4}{\pi} \]

    8. pow1/2 97.0%

       \[\leadsto \left(\left(0.25 \cdot \left({\left(\pi \cdot f\right)}^{2} \cdot \left({0.08333333333333333}^{0.5} \cdot \color{blue}{{0.08333333333333333}^{0.5}}\right)\right) - \left(\log f - \log 4\right)\right) + \left(-\log \pi\right)\right) \cdot \frac{-4}{\pi} \]

    9. pow-prod-up 97.0%

       \[\leadsto \left(\left(0.25 \cdot \left({\left(\pi \cdot f\right)}^{2} \cdot \color{blue}{{0.08333333333333333}^{\left(0.5 + 0.5\right)}}\right) - \left(\log f - \log 4\right)\right) + \left(-\log \pi\right)\right) \cdot \frac{-4}{\pi} \]

    10. metadata-eval 97.0%

        \[\leadsto \left(\left(0.25 \cdot \left({\left(\pi \cdot f\right)}^{2} \cdot {0.08333333333333333}^{\color{blue}{1}}\right) - \left(\log f - \log 4\right)\right) + \left(-\log \pi\right)\right) \cdot \frac{-4}{\pi} \]

    11. metadata-eval 97.0%

        \[\leadsto \left(\left(0.25 \cdot \left({\left(\pi \cdot f\right)}^{2} \cdot \color{blue}{0.08333333333333333}\right) - \left(\log f - \log 4\right)\right) + \left(-\log \pi\right)\right) \cdot \frac{-4}{\pi} \]

    12. diff-log 97.1%

        \[\leadsto \left(\left(0.25 \cdot \left({\left(\pi \cdot f\right)}^{2} \cdot 0.08333333333333333\right) - \color{blue}{\log \left(\frac{f}{4}\right)}\right) + \left(-\log \pi\right)\right) \cdot \frac{-4}{\pi} \]

11. Applied egg-rr 97.1%

    \[\leadsto \color{blue}{\left(\left(0.25 \cdot \left({\left(\pi \cdot f\right)}^{2} \cdot 0.08333333333333333\right) - \log \left(\frac{f}{4}\right)\right) + \left(-\log \pi\right)\right)} \cdot \frac{-4}{\pi} \]
12. Step-by-step derivation

    1. unsub-neg 97.1%

       \[\leadsto \color{blue}{\left(\left(0.25 \cdot \left({\left(\pi \cdot f\right)}^{2} \cdot 0.08333333333333333\right) - \log \left(\frac{f}{4}\right)\right) - \log \pi\right)} \cdot \frac{-4}{\pi} \]

    2. associate--l- 97.1%

       \[\leadsto \color{blue}{\left(0.25 \cdot \left({\left(\pi \cdot f\right)}^{2} \cdot 0.08333333333333333\right) - \left(\log \left(\frac{f}{4}\right) + \log \pi\right)\right)} \cdot \frac{-4}{\pi} \]

    3. *-commutative 97.1%

       \[\leadsto \left(\color{blue}{\left({\left(\pi \cdot f\right)}^{2} \cdot 0.08333333333333333\right) \cdot 0.25} - \left(\log \left(\frac{f}{4}\right) + \log \pi\right)\right) \cdot \frac{-4}{\pi} \]

    4. associate-*l* 97.1%

       \[\leadsto \left(\color{blue}{{\left(\pi \cdot f\right)}^{2} \cdot \left(0.08333333333333333 \cdot 0.25\right)} - \left(\log \left(\frac{f}{4}\right) + \log \pi\right)\right) \cdot \frac{-4}{\pi} \]

    5. metadata-eval 97.1%

       \[\leadsto \left({\left(\pi \cdot f\right)}^{2} \cdot \color{blue}{0.020833333333333332} - \left(\log \left(\frac{f}{4}\right) + \log \pi\right)\right) \cdot \frac{-4}{\pi} \]

    6. *-commutative 97.1%

       \[\leadsto \left({\color{blue}{\left(f \cdot \pi\right)}}^{2} \cdot 0.020833333333333332 - \left(\log \left(\frac{f}{4}\right) + \log \pi\right)\right) \cdot \frac{-4}{\pi} \]

13. Simplified 97.1%

    \[\leadsto \color{blue}{\left({\left(f \cdot \pi\right)}^{2} \cdot 0.020833333333333332 - \left(\log \left(\frac{f}{4}\right) + \log \pi\right)\right)} \cdot \frac{-4}{\pi} \]

14. Final simplification 97.1%

    \[\leadsto \frac{-4}{\pi} \cdot \left({\left(f \cdot \pi\right)}^{2} \cdot 0.020833333333333332 - \left(\log \left(\frac{f}{4}\right) + \log \pi\right)\right) \]
15. Add Preprocessing

Alternative 5: 95.9% 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 98.5%

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

5. Taylor expanded in f around 0 96.9%

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

   1. mul-1-neg 96.9%

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

   2. unsub-neg 96.9%

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

7. Simplified 96.9%

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

Alternative 6: 4.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_] := N[(-0.08333333333333333 * N[(Pi * N[Power[f, 2.0], $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 98.5%

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

5. Taylor expanded in f around 0 97.2%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + 0.25 \cdot \left({f}^{2} \cdot \left(\pi \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]
6. Step-by-step derivation

   1. +-commutative 97.2%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(0.25 \cdot \left({f}^{2} \cdot \left(\pi \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)\right) + -1 \cdot \log f\right)}\right) \cdot \frac{-4}{\pi} \]

   2. mul-1-neg 97.2%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \left(0.25 \cdot \left({f}^{2} \cdot \left(\pi \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)\right) + \color{blue}{\left(-\log f\right)}\right)\right) \cdot \frac{-4}{\pi} \]

7. Simplified 97.2%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(0.25 \cdot \left({f}^{2} \cdot \left(0.08333333333333333 \cdot {\pi}^{2}\right)\right) - \log f\right)\right)} \cdot \frac{-4}{\pi} \]

8. Taylor expanded in f around inf 4.2%

   \[\leadsto \color{blue}{\left(0.020833333333333332 \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right)} \cdot \frac{-4}{\pi} \]

9. Taylor expanded in f around 0 4.2%

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

10. Final simplification 4.2%

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

Reproduce

herbie shell --seed 2024103 
(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)))))))))