VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.2% → 98.9%

Time: 17.1s

Alternatives: 8

Speedup: 4.7×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 7.2% 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: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double f) {
	return -4.0 * (log1p(((1.0 / expm1((0.5 * (((double) M_PI) * f)))) + (-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((0.5 * (Math.PI * f)))) + (-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((0.5 * (math.pi * f)))) + (-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(Float64(0.5 * Float64(pi * f)))) + Float64(-1.0 + Float64(-1.0 / expm1(Float64(Float64(pi * f) * -0.5)))))) / pi))
end

Wolfram

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

Derivation

1. Initial program 7.8%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around inf 7.5%

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

   1. expm1-define 7.7%

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

   2. *-commutative 7.7%

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

   3. expm1-define 99.0%

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

   4. associate-*r* 99.0%

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

   5. *-commutative 99.0%

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

   6. *-commutative 99.0%

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

7. Simplified 99.0%

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

   1. log1p-expm1-u 99.0%

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

   2. expm1-undefine 99.0%

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

   3. add-exp-log 99.0%

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

9. Applied egg-rr 99.0%

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

    1. associate--l- 99.0%

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

    2. associate-*r* 99.0%

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

11. Simplified 99.0%

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

12. Final simplification 99.0%

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

Alternative 2: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double f) {
	return -4.0 * (log1p((((1.0 / expm1((0.5 * (((double) M_PI) * f)))) + -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((0.5 * (Math.PI * f)))) + -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((0.5 * (math.pi * f)))) + -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(Float64(1.0 / expm1(Float64(0.5 * Float64(pi * f)))) + -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[(N[(1.0 / N[(Exp[N[(0.5 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + N[(-1.0 / N[(Exp[N[(Pi * N[(f * -0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.8%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around inf 7.5%

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

   1. expm1-define 7.7%

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

   2. *-commutative 7.7%

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

   3. expm1-define 99.0%

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

   4. associate-*r* 99.0%

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

   5. *-commutative 99.0%

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

   6. *-commutative 99.0%

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

7. Simplified 99.0%

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

   1. log1p-expm1-u 99.0%

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

   2. expm1-undefine 99.0%

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

   3. add-exp-log 99.0%

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

9. Applied egg-rr 99.0%

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

    1. sub-neg 99.0%

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

    2. sub-neg 99.0%

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

    3. distribute-neg-frac 99.0%

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

    4. metadata-eval 99.0%

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

    5. +-commutative 99.0%

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

    6. metadata-eval 99.0%

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

    7. associate-+l+ 99.0%

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

11. Simplified 99.0%

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

12. Final simplification 99.0%

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

Alternative 3: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.8%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around inf 7.5%

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

   1. expm1-define 7.7%

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

   2. *-commutative 7.7%

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

   3. expm1-define 99.0%

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

   4. associate-*r* 99.0%

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

   5. *-commutative 99.0%

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

   6. *-commutative 99.0%

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

7. Simplified 99.0%

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

Alternative 4: 98.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;-4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} + \frac{\frac{2}{\pi} - f \cdot \left(0.5 + f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 225.0)
   (*
    -4.0
    (/
     (log1p
      (+
       (/ 1.0 (expm1 (* 0.5 (* PI f))))
       (/ (- (/ 2.0 PI) (* f (+ 0.5 (* f (* PI -0.041666666666666664))))) f)))
     PI))
   0.0))

C

double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = -4.0 * (log1p(((1.0 / expm1((0.5 * (((double) M_PI) * f)))) + (((2.0 / ((double) M_PI)) - (f * (0.5 + (f * (((double) M_PI) * -0.041666666666666664))))) / f))) / ((double) M_PI));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = -4.0 * (Math.log1p(((1.0 / Math.expm1((0.5 * (Math.PI * f)))) + (((2.0 / Math.PI) - (f * (0.5 + (f * (Math.PI * -0.041666666666666664))))) / f))) / Math.PI);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 225.0:
		tmp = -4.0 * (math.log1p(((1.0 / math.expm1((0.5 * (math.pi * f)))) + (((2.0 / math.pi) - (f * (0.5 + (f * (math.pi * -0.041666666666666664))))) / f))) / math.pi)
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 225.0)
		tmp = Float64(-4.0 * Float64(log1p(Float64(Float64(1.0 / expm1(Float64(0.5 * Float64(pi * f)))) + Float64(Float64(Float64(2.0 / pi) - Float64(f * Float64(0.5 + Float64(f * Float64(pi * -0.041666666666666664))))) / f))) / pi));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[f_] := If[LessEqual[f, 225.0], N[(-4.0 * N[(N[Log[1 + N[(N[(1.0 / N[(Exp[N[(0.5 * N[(Pi * f), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 / Pi), $MachinePrecision] - N[(f * N[(0.5 + N[(f * N[(Pi * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 225

   1. Initial program 8.0%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in f around inf 5.3%

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

      1. expm1-define 5.5%

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

      2. *-commutative 5.5%

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

      3. expm1-define 99.3%

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

      4. associate-*r* 99.3%

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

      5. *-commutative 99.3%

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

      6. *-commutative 99.3%

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

   7. Simplified 99.3%

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

      1. log1p-expm1-u 99.3%

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

      2. expm1-undefine 99.3%

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

      3. add-exp-log 99.3%

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

   9. Applied egg-rr 99.3%

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

       1. associate--l- 99.3%

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

       2. associate-*r* 99.3%

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

   11. Simplified 99.3%

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

   12. Taylor expanded in f around 0 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} - \color{blue}{\frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right)\right)\right) - 2 \cdot \frac{1}{\pi}}{f}}\right)}{\pi} \]
   13. Step-by-step derivation

       1. associate-*r* 98.9%

          \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} - \frac{f \cdot \left(0.5 + \color{blue}{\left(-1 \cdot f\right) \cdot \left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right)}\right) - 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

       2. mul-1-neg 98.9%

          \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} - \frac{f \cdot \left(0.5 + \color{blue}{\left(-f\right)} \cdot \left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right)\right) - 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

       3. distribute-rgt-out 98.9%

          \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} - \frac{f \cdot \left(0.5 + \left(-f\right) \cdot \color{blue}{\left(\pi \cdot \left(-0.08333333333333333 + 0.125\right)\right)}\right) - 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

       4. metadata-eval 98.9%

          \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} - \frac{f \cdot \left(0.5 + \left(-f\right) \cdot \left(\pi \cdot \color{blue}{0.041666666666666664}\right)\right) - 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

       5. associate-*r/ 98.9%

          \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} - \frac{f \cdot \left(0.5 + \left(-f\right) \cdot \left(\pi \cdot 0.041666666666666664\right)\right) - \color{blue}{\frac{2 \cdot 1}{\pi}}}{f}\right)}{\pi} \]

       6. metadata-eval 98.9%

          \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} - \frac{f \cdot \left(0.5 + \left(-f\right) \cdot \left(\pi \cdot 0.041666666666666664\right)\right) - \frac{\color{blue}{2}}{\pi}}{f}\right)}{\pi} \]

   14. Simplified 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} - \color{blue}{\frac{f \cdot \left(0.5 + \left(-f\right) \cdot \left(\pi \cdot 0.041666666666666664\right)\right) - \frac{2}{\pi}}{f}}\right)}{\pi} \]

   15. Taylor expanded in f around 0 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} - \frac{f \cdot \left(0.5 + \color{blue}{-0.041666666666666664 \cdot \left(f \cdot \pi\right)}\right) - \frac{2}{\pi}}{f}\right)}{\pi} \]
   16. Step-by-step derivation

       1. *-commutative 98.9%

          \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} - \frac{f \cdot \left(0.5 + \color{blue}{\left(f \cdot \pi\right) \cdot -0.041666666666666664}\right) - \frac{2}{\pi}}{f}\right)}{\pi} \]

       2. associate-*r* 98.9%

          \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} - \frac{f \cdot \left(0.5 + \color{blue}{f \cdot \left(\pi \cdot -0.041666666666666664\right)}\right) - \frac{2}{\pi}}{f}\right)}{\pi} \]

   17. Simplified 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} - \frac{f \cdot \left(0.5 + \color{blue}{f \cdot \left(\pi \cdot -0.041666666666666664\right)}\right) - \frac{2}{\pi}}{f}\right)}{\pi} \]

   if 225 < f

   1. Initial program 1.0%

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

   3. Simplified 86.7%

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

   6. Applied egg-rr 3.2%

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

      1. unpow2 3.2%

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

      2. add-sqr-sqrt 3.2%

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

      3. flip-+ 0.0%

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

      4. log-div 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

         \[\leadsto \left(\log \color{blue}{0} - \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} - \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\right) \cdot \frac{-4}{\pi} \]

      2. +-inverses 0.0%

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

      3. +-inverses 86.7%

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

   10. Simplified 86.7%

       \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]
   11. Step-by-step derivation

       1. mul0-lft 86.7%

          \[\leadsto \color{blue}{0} \]

   12. Applied egg-rr 86.7%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;-4 \cdot \frac{\mathsf{log1p}\left(\frac{1}{\mathsf{expm1}\left(0.5 \cdot \left(\pi \cdot f\right)\right)} + \frac{\frac{2}{\pi} - f \cdot \left(0.5 + f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;-4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(-1 + \pi \cdot \left(f \cdot 0.08333333333333333\right)\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 225.0)
   (*
    -4.0
    (/
     (log1p
      (/
       (+ (* f (+ -1.0 (* PI (* f 0.08333333333333333)))) (* 4.0 (/ 1.0 PI)))
       f))
     PI))
   0.0))

C

double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = -4.0 * (log1p((((f * (-1.0 + (((double) M_PI) * (f * 0.08333333333333333)))) + (4.0 * (1.0 / ((double) M_PI)))) / f)) / ((double) M_PI));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 225.0) {
		tmp = -4.0 * (Math.log1p((((f * (-1.0 + (Math.PI * (f * 0.08333333333333333)))) + (4.0 * (1.0 / Math.PI))) / f)) / Math.PI);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 225.0:
		tmp = -4.0 * (math.log1p((((f * (-1.0 + (math.pi * (f * 0.08333333333333333)))) + (4.0 * (1.0 / math.pi))) / f)) / math.pi)
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 225.0)
		tmp = Float64(-4.0 * Float64(log1p(Float64(Float64(Float64(f * Float64(-1.0 + Float64(pi * Float64(f * 0.08333333333333333)))) + Float64(4.0 * Float64(1.0 / pi))) / f)) / pi));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[f_] := If[LessEqual[f, 225.0], N[(-4.0 * N[(N[Log[1 + N[(N[(N[(f * N[(-1.0 + N[(Pi * N[(f * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 225

   1. Initial program 8.0%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in f around inf 5.3%

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

      1. expm1-define 5.5%

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

      2. *-commutative 5.5%

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

      3. expm1-define 99.3%

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

      4. associate-*r* 99.3%

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

      5. *-commutative 99.3%

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

      6. *-commutative 99.3%

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

   7. Simplified 99.3%

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

      1. log1p-expm1-u 99.3%

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

      2. expm1-undefine 99.3%

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

      3. add-exp-log 99.3%

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

   9. Applied egg-rr 99.3%

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

       1. associate--l- 99.3%

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

       2. associate-*r* 99.3%

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

   11. Simplified 99.3%

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

   12. Taylor expanded in f around 0 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\color{blue}{\frac{f \cdot \left(f \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right) - 1\right) + 4 \cdot \frac{1}{\pi}}{f}}\right)}{\pi} \]
   13. Step-by-step derivation

       1. pow1 98.9%

          \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(\color{blue}{{\left(f \cdot \left(\left(-0.08333333333333333 \cdot \pi + 0.125 \cdot \pi\right) - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)}^{1}} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

       2. distribute-rgt-out 98.9%

          \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left({\left(f \cdot \left(\color{blue}{\pi \cdot \left(-0.08333333333333333 + 0.125\right)} - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)}^{1} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

       3. metadata-eval 98.9%

          \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left({\left(f \cdot \left(\pi \cdot \color{blue}{0.041666666666666664} - \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right)\right)}^{1} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

       4. distribute-rgt-out 98.9%

          \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left({\left(f \cdot \left(\pi \cdot 0.041666666666666664 - \color{blue}{\pi \cdot \left(-0.125 + 0.08333333333333333\right)}\right)\right)}^{1} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

       5. metadata-eval 98.9%

          \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left({\left(f \cdot \left(\pi \cdot 0.041666666666666664 - \pi \cdot \color{blue}{-0.041666666666666664}\right)\right)}^{1} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

   14. Applied egg-rr 98.9%

       \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(\color{blue}{{\left(f \cdot \left(\pi \cdot 0.041666666666666664 - \pi \cdot -0.041666666666666664\right)\right)}^{1}} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]
   15. Step-by-step derivation

       1. unpow1 98.9%

          \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(\color{blue}{f \cdot \left(\pi \cdot 0.041666666666666664 - \pi \cdot -0.041666666666666664\right)} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

       2. *-commutative 98.9%

          \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(\color{blue}{\left(\pi \cdot 0.041666666666666664 - \pi \cdot -0.041666666666666664\right) \cdot f} - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

       3. distribute-lft-out-- 98.9%

          \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(\color{blue}{\left(\pi \cdot \left(0.041666666666666664 - -0.041666666666666664\right)\right)} \cdot f - 1\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

       4. metadata-eval 98.9%

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

       5. associate-*l* 98.9%

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

   16. Simplified 98.9%

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

   if 225 < f

   1. Initial program 1.0%

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

   3. Simplified 86.7%

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

   6. Applied egg-rr 3.2%

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

      1. unpow2 3.2%

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

      2. add-sqr-sqrt 3.2%

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

      3. flip-+ 0.0%

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

      4. log-div 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

         \[\leadsto \left(\log \color{blue}{0} - \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} - \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\right) \cdot \frac{-4}{\pi} \]

      2. +-inverses 0.0%

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

      3. +-inverses 86.7%

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

   10. Simplified 86.7%

       \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]
   11. Step-by-step derivation

       1. mul0-lft 86.7%

          \[\leadsto \color{blue}{0} \]

   12. Applied egg-rr 86.7%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;-4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(-1 + \pi \cdot \left(f \cdot 0.08333333333333333\right)\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;-4 \cdot \frac{\mathsf{log1p}\left(\frac{\frac{4}{\pi} - f}{f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.26) (* -4.0 (/ (log1p (/ (- (/ 4.0 PI) f) f)) PI)) 0.0))

C

double code(double f) {
	double tmp;
	if (f <= 1.26) {
		tmp = -4.0 * (log1p((((4.0 / ((double) M_PI)) - f) / f)) / ((double) M_PI));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.26) {
		tmp = -4.0 * (Math.log1p((((4.0 / Math.PI) - f) / f)) / Math.PI);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.26:
		tmp = -4.0 * (math.log1p((((4.0 / math.pi) - f) / f)) / math.pi)
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.26)
		tmp = Float64(-4.0 * Float64(log1p(Float64(Float64(Float64(4.0 / pi) - f) / f)) / pi));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[f_] := If[LessEqual[f, 1.26], N[(-4.0 * N[(N[Log[1 + N[(N[(N[(4.0 / Pi), $MachinePrecision] - f), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 1.26000000000000001

   1. Initial program 7.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in f around inf 5.0%

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

      1. expm1-define 5.3%

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

      2. *-commutative 5.3%

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

      3. expm1-define 99.5%

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

      4. associate-*r* 99.5%

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

      5. *-commutative 99.5%

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

      6. *-commutative 99.5%

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

   7. Simplified 99.5%

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

      1. log1p-expm1-u 99.5%

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

      2. expm1-undefine 99.5%

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

      3. add-exp-log 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. associate--l- 99.5%

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

       2. associate-*r* 99.5%

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

   11. Simplified 99.5%

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

   12. Taylor expanded in f around 0 98.4%

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

       1. +-commutative 98.4%

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

       2. mul-1-neg 98.4%

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

       3. unsub-neg 98.4%

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

       4. associate-*r/ 98.4%

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

       5. metadata-eval 98.4%

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

   14. Simplified 98.4%

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

   if 1.26000000000000001 < f

   1. Initial program 8.6%

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

   3. Simplified 83.6%

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

   6. Applied egg-rr 4.2%

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

      1. unpow2 4.2%

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

      2. add-sqr-sqrt 4.2%

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

      3. flip-+ 0.0%

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

      4. log-div 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

         \[\leadsto \left(\log \color{blue}{0} - \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} - \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\right) \cdot \frac{-4}{\pi} \]

      2. +-inverses 0.0%

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

      3. +-inverses 76.3%

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

   10. Simplified 76.3%

       \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]
   11. Step-by-step derivation

       1. mul0-lft 76.3%

          \[\leadsto \color{blue}{0} \]

   12. Applied egg-rr 76.3%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq 1.26:\\ \;\;\;\;-4 \cdot \frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (if (<= f 1.26) (* -4.0 (/ (log (/ 4.0 (* PI f))) PI)) 0.0))

C

double code(double f) {
	double tmp;
	if (f <= 1.26) {
		tmp = -4.0 * (log((4.0 / (((double) M_PI) * f))) / ((double) M_PI));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double f) {
	double tmp;
	if (f <= 1.26) {
		tmp = -4.0 * (Math.log((4.0 / (Math.PI * f))) / Math.PI);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(f):
	tmp = 0
	if f <= 1.26:
		tmp = -4.0 * (math.log((4.0 / (math.pi * f))) / math.pi)
	else:
		tmp = 0.0
	return tmp

Julia

function code(f)
	tmp = 0.0
	if (f <= 1.26)
		tmp = Float64(-4.0 * Float64(log(Float64(4.0 / Float64(pi * f))) / pi));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f)
	tmp = 0.0;
	if (f <= 1.26)
		tmp = -4.0 * (log((4.0 / (pi * f))) / pi);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_] := If[LessEqual[f, 1.26], N[(-4.0 * N[(N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if f < 1.26000000000000001

   1. Initial program 7.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in f around inf 5.0%

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

      1. expm1-define 5.3%

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

      2. *-commutative 5.3%

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

      3. expm1-define 99.5%

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

      4. associate-*r* 99.5%

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

      5. *-commutative 99.5%

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

      6. *-commutative 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in f around 0 98.4%

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

      1. *-commutative 98.4%

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

   10. Simplified 98.4%

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

   if 1.26000000000000001 < f

   1. Initial program 8.6%

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

   3. Simplified 83.6%

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

   6. Applied egg-rr 4.2%

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

      1. unpow2 4.2%

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

      2. add-sqr-sqrt 4.2%

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

      3. flip-+ 0.0%

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

      4. log-div 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. +-inverses 0.0%

         \[\leadsto \left(\log \color{blue}{0} - \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} - \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\right) \cdot \frac{-4}{\pi} \]

      2. +-inverses 0.0%

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

      3. +-inverses 76.3%

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

   10. Simplified 76.3%

       \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]
   11. Step-by-step derivation

       1. mul0-lft 76.3%

          \[\leadsto \color{blue}{0} \]

   12. Applied egg-rr 76.3%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 5.3% accurate, 532.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (f) :precision binary64 0.0)

C

double code(double f) {
	return 0.0;
}

Fortran

real(8) function code(f)
    real(8), intent (in) :: f
    code = 0.0d0
end function

Java

public static double code(double f) {
	return 0.0;
}

Python

def code(f):
	return 0.0

Julia

function code(f)
	return 0.0
end

MATLAB

function tmp = code(f)
	tmp = 0.0;
end

Wolfram

code[f_] := 0.0

Derivation

1. Initial program 7.8%

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

3. Simplified 98.9%

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

6. Applied egg-rr 93.6%

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

   1. unpow2 93.6%

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

   2. add-sqr-sqrt 94.0%

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

   3. flip-+ 0.0%

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

   4. log-div 0.0%

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

8. Applied egg-rr 0.0%

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

   1. +-inverses 0.0%

      \[\leadsto \left(\log \color{blue}{0} - \log \left(\frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)} - \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right)\right) \cdot \frac{-4}{\pi} \]

   2. +-inverses 0.0%

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

   3. +-inverses 5.4%

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

10. Simplified 5.4%

    \[\leadsto \color{blue}{0} \cdot \frac{-4}{\pi} \]
11. Step-by-step derivation

    1. mul0-lft 5.4%

       \[\leadsto \color{blue}{0} \]

12. Applied egg-rr 5.4%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024160 
(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)))))))))