VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 99.0%

Time: 17.8s

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: 6.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double f) {
	return -4.0 * (log1p(((1.0 / expm1((((double) M_PI) * (0.5 * 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((Math.PI * (0.5 * 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((math.pi * (0.5 * 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(pi * Float64(0.5 * f)))) + 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[(Pi * N[(0.5 * f), $MachinePrecision]), $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 7.2%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around inf 5.2%

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

7. Simplified 99.2%

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

   1. log1p-expm1-u 99.2%

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

   2. expm1-undefine 99.2%

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

   3. add-exp-log 99.2%

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

   4. associate-*r* 99.2%

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

   5. *-commutative 99.2%

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

   6. *-commutative 99.2%

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

   7. *-commutative 99.2%

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

   8. associate-*l* 99.2%

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

   9. *-commutative 99.2%

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

9. Applied egg-rr 99.2%

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

    1. associate--l+ 99.2%

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

    2. expm1-undefine 8.1%

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

    3. *-commutative 8.1%

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

    4. *-commutative 8.1%

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

    5. expm1-define 99.2%

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

    6. *-commutative 99.2%

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

    7. *-commutative 99.2%

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

    8. associate-*l* 99.2%

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

    9. *-commutative 99.2%

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

    10. sub-neg 99.2%

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

11. Simplified 99.2%

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

12. Final simplification 99.2%

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

Alternative 2: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Python

def code(f):
	return -4.0 * (math.log(((1.0 / math.expm1((f * (math.pi * 0.5)))) + (-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(f * Float64(pi * 0.5)))) + 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[(f * N[(Pi * 0.5), $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 7.2%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around inf 5.2%

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

7. Simplified 99.2%

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

Alternative 3: 96.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\mathsf{log1p}\left(\left(-1 + \frac{-1}{\mathsf{expm1}\left(\pi \cdot \left(f \cdot -0.5\right)\right)}\right) + \frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)}{f}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log1p
    (+
     (+ -1.0 (/ -1.0 (expm1 (* PI (* f -0.5)))))
     (/
      (-
       (* 2.0 (/ 1.0 PI))
       (* f (+ 0.5 (* f (+ (* PI -0.125) (* PI 0.08333333333333333))))))
      f)))
   PI)))

C

double code(double f) {
	return -4.0 * (log1p(((-1.0 + (-1.0 / expm1((((double) M_PI) * (f * -0.5))))) + (((2.0 * (1.0 / ((double) M_PI))) - (f * (0.5 + (f * ((((double) M_PI) * -0.125) + (((double) M_PI) * 0.08333333333333333)))))) / f))) / ((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))))) + (((2.0 * (1.0 / Math.PI)) - (f * (0.5 + (f * ((Math.PI * -0.125) + (Math.PI * 0.08333333333333333)))))) / f))) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log1p(((-1.0 + (-1.0 / math.expm1((math.pi * (f * -0.5))))) + (((2.0 * (1.0 / math.pi)) - (f * (0.5 + (f * ((math.pi * -0.125) + (math.pi * 0.08333333333333333)))))) / f))) / 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(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))) / 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[(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]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.2%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around inf 5.2%

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

7. Simplified 99.2%

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

   1. log1p-expm1-u 99.2%

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

   2. expm1-undefine 99.2%

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

   3. add-exp-log 99.2%

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

   4. associate-*r* 99.2%

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

   5. *-commutative 99.2%

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

   6. *-commutative 99.2%

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

   7. *-commutative 99.2%

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

   8. associate-*l* 99.2%

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

   9. *-commutative 99.2%

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

9. Applied egg-rr 99.2%

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

    1. associate--l+ 99.2%

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

    2. expm1-undefine 8.1%

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

    3. *-commutative 8.1%

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

    4. *-commutative 8.1%

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

    5. expm1-define 99.2%

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

    6. *-commutative 99.2%

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

    7. *-commutative 99.2%

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

    8. associate-*l* 99.2%

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

    9. *-commutative 99.2%

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

    10. sub-neg 99.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in f around 0 97.4%

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

13. Final simplification 97.4%

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

Alternative 4: 96.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(-1 + f \cdot \left(\pi \cdot 0.08333333333333333\right)\right) + \frac{1}{\pi} \cdot 4}{f}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  -4.0
  (/
   (log1p
    (/
     (+ (* f (+ -1.0 (* f (* PI 0.08333333333333333)))) (* (/ 1.0 PI) 4.0))
     f))
   PI)))

C

double code(double f) {
	return -4.0 * (log1p((((f * (-1.0 + (f * (((double) M_PI) * 0.08333333333333333)))) + ((1.0 / ((double) M_PI)) * 4.0)) / f)) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log1p((((f * (-1.0 + (f * (Math.PI * 0.08333333333333333)))) + ((1.0 / Math.PI) * 4.0)) / f)) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log1p((((f * (-1.0 + (f * (math.pi * 0.08333333333333333)))) + ((1.0 / math.pi) * 4.0)) / f)) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log1p(Float64(Float64(Float64(f * Float64(-1.0 + Float64(f * Float64(pi * 0.08333333333333333)))) + Float64(Float64(1.0 / pi) * 4.0)) / f)) / pi))
end

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around inf 5.2%

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

7. Simplified 99.2%

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

   1. log1p-expm1-u 99.2%

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

   2. expm1-undefine 99.2%

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

   3. add-exp-log 99.2%

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

   4. associate-*r* 99.2%

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

   5. *-commutative 99.2%

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

   6. *-commutative 99.2%

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

   7. *-commutative 99.2%

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

   8. associate-*l* 99.2%

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

   9. *-commutative 99.2%

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

9. Applied egg-rr 99.2%

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

    1. associate--l+ 99.2%

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

    2. expm1-undefine 8.1%

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

    3. *-commutative 8.1%

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

    4. *-commutative 8.1%

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

    5. expm1-define 99.2%

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

    6. *-commutative 99.2%

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

    7. *-commutative 99.2%

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

    8. associate-*l* 99.2%

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

    9. *-commutative 99.2%

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

    10. sub-neg 99.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in f around 0 97.4%

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

    1. pow1 97.4%

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

    2. distribute-rgt-out 97.4%

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

    3. metadata-eval 97.4%

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

    4. distribute-rgt-out 97.4%

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

    5. metadata-eval 97.4%

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

14. Applied egg-rr 97.4%

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

    1. unpow1 97.4%

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

    2. distribute-lft-out-- 97.4%

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

    3. metadata-eval 97.4%

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

16. Simplified 97.4%

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

17. Final simplification 97.4%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{f \cdot \left(-1 + f \cdot \left(\pi \cdot 0.08333333333333333\right)\right) + \frac{1}{\pi} \cdot 4}{f}\right)}{\pi} \]
18. Add Preprocessing

Alternative 5: 96.1% accurate, 4.9× speedup

Math

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

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

Derivation

1. Initial program 7.2%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around 0 97.0%

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

   1. mul-1-neg 97.0%

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

   2. unsub-neg 97.0%

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

7. Simplified 97.0%

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

   1. associate-*r/ 97.0%

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

   2. diff-log 96.9%

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

9. Applied egg-rr 96.9%

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

Alternative 6: 96.0% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around 0 96.8%

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

   1. *-commutative 96.8%

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

   2. associate-/r* 96.8%

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

7. Simplified 96.8%

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

Alternative 7: 95.3% accurate, 4.9× speedup

Math

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

FPCore

(FPCore (f) :precision binary64 (* -4.0 (/ (log1p (/ 4.0 (* PI f))) PI)))

C

double code(double f) {
	return -4.0 * (log1p((4.0 / (((double) M_PI) * f))) / ((double) M_PI));
}

Java

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

Python

def code(f):
	return -4.0 * (math.log1p((4.0 / (math.pi * f))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log1p(Float64(4.0 / Float64(pi * f))) / pi))
end

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around inf 5.2%

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

7. Simplified 99.2%

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

   1. log1p-expm1-u 99.2%

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

   2. expm1-undefine 99.2%

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

   3. add-exp-log 99.2%

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

   4. associate-*r* 99.2%

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

   5. *-commutative 99.2%

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

   6. *-commutative 99.2%

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

   7. *-commutative 99.2%

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

   8. associate-*l* 99.2%

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

   9. *-commutative 99.2%

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

9. Applied egg-rr 99.2%

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

    1. associate--l+ 99.2%

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

    2. expm1-undefine 8.1%

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

    3. *-commutative 8.1%

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

    4. *-commutative 8.1%

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

    5. expm1-define 99.2%

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

    6. *-commutative 99.2%

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

    7. *-commutative 99.2%

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

    8. associate-*l* 99.2%

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

    9. *-commutative 99.2%

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

    10. sub-neg 99.2%

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

11. Simplified 99.2%

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

    1. add-cbrt-cube 99.2%

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

    2. pow3 99.2%

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

13. Applied egg-rr 99.2%

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

14. Taylor expanded in f around 0 96.2%

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

15. Final simplification 96.2%

    \[\leadsto -4 \cdot \frac{\mathsf{log1p}\left(\frac{4}{\pi \cdot f}\right)}{\pi} \]
16. Add Preprocessing

Alternative 8: 0.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \frac{-4}{\pi} \cdot \log 0 \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ -4.0 PI) (log 0.0)))

C

double code(double f) {
	return (-4.0 / ((double) M_PI)) * log(0.0);
}

Java

public static double code(double f) {
	return (-4.0 / Math.PI) * Math.log(0.0);
}

Python

def code(f):
	return (-4.0 / math.pi) * math.log(0.0)

Julia

function code(f)
	return Float64(Float64(-4.0 / pi) * log(0.0))
end

MATLAB

function tmp = code(f)
	tmp = (-4.0 / pi) * log(0.0);
end

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Simplified 99.1%

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

5. Taylor expanded in f around inf 5.2%

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

   1. expm1-define 5.4%

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

   2. metadata-eval 5.4%

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

   3. *-commutative 5.4%

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

   4. rem-square-sqrt 5.3%

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

   5. fabs-sqr 5.3%

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

   6. rem-square-sqrt 5.4%

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

   7. fabs-mul 5.4%

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

   8. *-commutative 5.4%

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

   9. associate-*r* 5.4%

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

   10. rem-square-sqrt 0.0%

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

   11. fabs-sqr 0.0%

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

   12. rem-square-sqrt 0.3%

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

   13. expm1-define 0.7%

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

7. Simplified 0.7%

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

8. Final simplification 0.7%

   \[\leadsto \frac{-4}{\pi} \cdot \log 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024135 
(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)))))))))