VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 99.0%

Time: 32.5s

Alternatives: 8

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: 6.9% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.2%

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

3. Simplified 98.4%

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

5. Taylor expanded in f around inf 6.5%

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

   1. expm1-define 6.7%

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

   2. *-commutative 6.7%

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

   3. expm1-define 98.6%

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

   4. *-commutative 98.6%

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

7. Simplified 98.6%

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

   1. log1p-expm1-u 98.6%

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

   2. expm1-undefine 98.6%

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

   3. add-exp-log 98.6%

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

   4. associate-*r* 98.6%

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

   5. *-commutative 98.6%

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

   6. associate-*l* 98.6%

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

   7. *-commutative 98.6%

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

   8. *-commutative 98.6%

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

9. Applied egg-rr 98.6%

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

    1. sub-neg 98.6%

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

    2. sub-neg 98.6%

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

    3. metadata-eval 98.6%

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

    4. associate-+l+ 98.6%

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

    5. distribute-neg-frac 98.6%

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

    6. metadata-eval 98.6%

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

11. Simplified 98.6%

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

    1. expm1-log1p-u 98.6%

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

    2. expm1-undefine 9.5%

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

13. Applied egg-rr 9.5%

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

    1. expm1-define 98.6%

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

15. Simplified 98.6%

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

16. Final simplification 98.6%

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

Alternative 2: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.2%

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

3. Simplified 98.4%

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

5. Taylor expanded in f around inf 6.5%

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

   1. expm1-define 6.7%

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

   2. *-commutative 6.7%

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

   3. expm1-define 98.6%

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

   4. *-commutative 98.6%

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

7. Simplified 98.6%

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

   1. log1p-expm1-u 98.6%

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

   2. expm1-undefine 98.6%

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

   3. add-exp-log 98.6%

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

   4. associate-*r* 98.6%

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

   5. *-commutative 98.6%

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

   6. associate-*l* 98.6%

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

   7. *-commutative 98.6%

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

   8. *-commutative 98.6%

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

9. Applied egg-rr 98.6%

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

    1. sub-neg 98.6%

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

    2. sub-neg 98.6%

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

    3. metadata-eval 98.6%

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

    4. associate-+l+ 98.6%

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

    5. distribute-neg-frac 98.6%

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

    6. metadata-eval 98.6%

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

11. Simplified 98.6%

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

12. Final simplification 98.6%

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

Alternative 3: 98.9% accurate, 1.7× speedup

Math

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

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(-4.0 * Float64(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[(-4.0 * 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[(-0.5 * N[(f * Pi), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.2%

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

3. Simplified 98.4%

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

5. Taylor expanded in f around inf 6.5%

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

   1. expm1-define 6.7%

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

   2. *-commutative 6.7%

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

   3. expm1-define 98.6%

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

   4. *-commutative 98.6%

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

7. Simplified 98.6%

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

8. Final simplification 98.6%

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

Alternative 4: 96.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)}{f} + \frac{-1}{\mathsf{expm1}\left(-0.5 \cdot \left(f \cdot \pi\right)\right)}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log
    (+
     (/
      (-
       (* 2.0 (/ 1.0 PI))
       (* f (+ 0.5 (* f (+ (* PI -0.125) (* PI 0.08333333333333333))))))
      f)
     (/ -1.0 (expm1 (* -0.5 (* f PI))))))
   PI)))

C

double code(double f) {
	return -4.0 * (log(((((2.0 * (1.0 / ((double) M_PI))) - (f * (0.5 + (f * ((((double) M_PI) * -0.125) + (((double) M_PI) * 0.08333333333333333)))))) / f) + (-1.0 / expm1((-0.5 * (f * ((double) M_PI))))))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log(((((2.0 * (1.0 / Math.PI)) - (f * (0.5 + (f * ((Math.PI * -0.125) + (Math.PI * 0.08333333333333333)))))) / f) + (-1.0 / Math.expm1((-0.5 * (f * Math.PI)))))) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log(((((2.0 * (1.0 / math.pi)) - (f * (0.5 + (f * ((math.pi * -0.125) + (math.pi * 0.08333333333333333)))))) / f) + (-1.0 / math.expm1((-0.5 * (f * math.pi)))))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log(Float64(Float64(Float64(Float64(2.0 * Float64(1.0 / pi)) - Float64(f * Float64(0.5 + Float64(f * Float64(Float64(pi * -0.125) + Float64(pi * 0.08333333333333333)))))) / f) + Float64(-1.0 / expm1(Float64(-0.5 * Float64(f * pi)))))) / pi))
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(N[(N[(N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] - N[(f * N[(0.5 + N[(f * N[(N[(Pi * -0.125), $MachinePrecision] + N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision] + N[(-1.0 / N[(Exp[N[(-0.5 * N[(f * Pi), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.2%

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

3. Simplified 98.4%

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

5. Taylor expanded in f around inf 6.5%

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

   1. expm1-define 6.7%

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

   2. *-commutative 6.7%

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

   3. expm1-define 98.6%

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

   4. *-commutative 98.6%

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

7. Simplified 98.6%

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

8. Taylor expanded in f around 0 96.3%

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

9. Final simplification 96.3%

   \[\leadsto -4 \cdot \frac{\log \left(\frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)}{f} + \frac{-1}{\mathsf{expm1}\left(-0.5 \cdot \left(f \cdot \pi\right)\right)}\right)}{\pi} \]
10. Add Preprocessing

Alternative 5: 96.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.2%

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

3. Simplified 98.4%

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

5. Taylor expanded in f around 0 96.1%

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

   1. *-un-lft-identity 96.1%

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

   2. distribute-rgt-out 96.1%

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

   3. metadata-eval 96.1%

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

7. Applied egg-rr 96.1%

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

   1. *-lft-identity 96.1%

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

9. Simplified 96.1%

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

10. Final simplification 96.1%

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

Alternative 6: 95.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{-\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* -4.0 (/ (log (* f (* PI 0.25))) (- PI))))

C

double code(double f) {
	return -4.0 * (log((f * (((double) M_PI) * 0.25))) / -((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log((f * (Math.PI * 0.25))) / -Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log((f * (math.pi * 0.25))) / -math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log(Float64(f * Float64(pi * 0.25))) / Float64(-pi)))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 * (log((f * (pi * 0.25))) / -pi);
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(f * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-Pi)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.2%

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

3. Simplified 98.4%

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

5. Taylor expanded in f around 0 95.7%

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

   1. mul-1-neg 95.7%

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

   2. unsub-neg 95.7%

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

7. Simplified 95.7%

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

   1. diff-log 95.4%

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

9. Applied egg-rr 95.4%

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

    1. clear-num 95.4%

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

    2. log-div 95.8%

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

    3. metadata-eval 95.8%

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

    4. div-inv 95.8%

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

    5. clear-num 95.8%

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

    6. div-inv 95.8%

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

    7. metadata-eval 95.8%

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

11. Applied egg-rr 95.8%

    \[\leadsto -4 \cdot \frac{\color{blue}{0 - \log \left(f \cdot \left(\pi \cdot 0.25\right)\right)}}{\pi} \]
12. Step-by-step derivation

    1. neg-sub0 95.8%

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

13. Simplified 95.8%

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

14. Final simplification 95.8%

    \[\leadsto -4 \cdot \frac{\log \left(f \cdot \left(\pi \cdot 0.25\right)\right)}{-\pi} \]
15. Add Preprocessing

Alternative 7: 1.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\log \left(\pi \cdot \left(f \cdot 0.25\right)\right)}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* -4.0 (/ (log (* PI (* f 0.25))) PI)))

C

double code(double f) {
	return -4.0 * (log((((double) M_PI) * (f * 0.25))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log((Math.PI * (f * 0.25))) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log((math.pi * (f * 0.25))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log(Float64(pi * Float64(f * 0.25))) / pi))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 * (log((pi * (f * 0.25))) / pi);
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(Pi * N[(f * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.2%

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

3. Simplified 98.4%

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

5. Taylor expanded in f around 0 95.7%

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

   1. mul-1-neg 95.7%

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

   2. unsub-neg 95.7%

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

7. Simplified 95.7%

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

   1. diff-log 95.4%

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

9. Applied egg-rr 95.4%

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

    1. clear-num 95.4%

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

    2. log-div 95.8%

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

    3. metadata-eval 95.8%

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

    4. div-inv 95.8%

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

    5. clear-num 95.8%

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

    6. div-inv 95.8%

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

    7. metadata-eval 95.8%

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

11. Applied egg-rr 95.8%

    \[\leadsto -4 \cdot \frac{\color{blue}{0 - \log \left(f \cdot \left(\pi \cdot 0.25\right)\right)}}{\pi} \]
12. Step-by-step derivation

    1. neg-sub0 95.8%

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

13. Simplified 95.8%

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

    1. *-un-lft-identity 95.8%

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

    2. metadata-eval 95.8%

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

    3. times-frac 95.8%

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

    4. add-sqr-sqrt 95.3%

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

    5. sqrt-unprod 96.0%

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

    6. sqr-neg 96.0%

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

    7. sqrt-unprod 0.3%

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

    8. add-sqr-sqrt 1.8%

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

    9. times-frac 1.8%

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

    10. metadata-eval 1.8%

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

15. Applied egg-rr 1.8%

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

    1. *-lft-identity 1.8%

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

    2. *-commutative 1.8%

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

    3. associate-*l* 1.8%

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

17. Simplified 1.8%

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

18. Final simplification 1.8%

    \[\leadsto -4 \cdot \frac{\log \left(\pi \cdot \left(f \cdot 0.25\right)\right)}{\pi} \]
19. Add Preprocessing

Alternative 8: 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 8.2%

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

3. Simplified 98.4%

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

5. Taylor expanded in f around 0 95.7%

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

   1. mul-1-neg 95.7%

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

   2. unsub-neg 95.7%

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

7. Simplified 95.7%

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

   1. diff-log 95.4%

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

9. Applied egg-rr 95.4%

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

10. Final simplification 95.4%

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

Reproduce

herbie shell --seed 2024096 
(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)))))))))