VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.0%

Time: 43.2s

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: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.2%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around inf 5.4%

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

   1. associate-*r/ 5.4%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

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

Alternative 2: 98.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.2%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 3: 96.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.2%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around inf 5.4%

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

   1. associate-*r/ 5.4%

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

7. Simplified 99.0%

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

8. Taylor expanded in f around 0 96.3%

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

   1. *-un-lft-identity 96.3%

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

   2. distribute-rgt-out 96.3%

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

   3. metadata-eval 96.3%

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

10. Applied egg-rr 96.3%

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

    1. *-lft-identity 96.3%

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

12. Simplified 96.3%

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

13. Taylor expanded in f around 0 96.3%

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

14. Final simplification 96.3%

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

Alternative 4: 96.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \frac{1}{\pi}\\ \frac{-4}{\pi} \cdot \log \left(\frac{t\_0 - f \cdot \left(0.5 + f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)}{f} + \frac{t\_0 + f \cdot \left(0.5 - f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)}{f}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* 2.0 (/ 1.0 PI))))
   (*
    (/ -4.0 PI)
    (log
     (+
      (/
       (- t_0 (* f (+ 0.5 (* f (+ (* PI -0.125) (* PI 0.08333333333333333))))))
       f)
      (/ (+ t_0 (* f (- 0.5 (* f (* PI -0.041666666666666664))))) f))))))

C

double code(double f) {
	double t_0 = 2.0 * (1.0 / ((double) M_PI));
	return (-4.0 / ((double) M_PI)) * log((((t_0 - (f * (0.5 + (f * ((((double) M_PI) * -0.125) + (((double) M_PI) * 0.08333333333333333)))))) / f) + ((t_0 + (f * (0.5 - (f * (((double) M_PI) * -0.041666666666666664))))) / f)));
}

Java

public static double code(double f) {
	double t_0 = 2.0 * (1.0 / Math.PI);
	return (-4.0 / Math.PI) * Math.log((((t_0 - (f * (0.5 + (f * ((Math.PI * -0.125) + (Math.PI * 0.08333333333333333)))))) / f) + ((t_0 + (f * (0.5 - (f * (Math.PI * -0.041666666666666664))))) / f)));
}

Python

def code(f):
	t_0 = 2.0 * (1.0 / math.pi)
	return (-4.0 / math.pi) * math.log((((t_0 - (f * (0.5 + (f * ((math.pi * -0.125) + (math.pi * 0.08333333333333333)))))) / f) + ((t_0 + (f * (0.5 - (f * (math.pi * -0.041666666666666664))))) / f)))

Julia

function code(f)
	t_0 = Float64(2.0 * Float64(1.0 / pi))
	return Float64(Float64(-4.0 / pi) * log(Float64(Float64(Float64(t_0 - Float64(f * Float64(0.5 + Float64(f * Float64(Float64(pi * -0.125) + Float64(pi * 0.08333333333333333)))))) / f) + Float64(Float64(t_0 + Float64(f * Float64(0.5 - Float64(f * Float64(pi * -0.041666666666666664))))) / f))))
end

MATLAB

function tmp = code(f)
	t_0 = 2.0 * (1.0 / pi);
	tmp = (-4.0 / pi) * log((((t_0 - (f * (0.5 + (f * ((pi * -0.125) + (pi * 0.08333333333333333)))))) / f) + ((t_0 + (f * (0.5 - (f * (pi * -0.041666666666666664))))) / f)));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[(N[(N[(t$95$0 - N[(f * N[(0.5 + N[(f * N[(N[(Pi * -0.125), $MachinePrecision] + N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision] + N[(N[(t$95$0 + N[(f * N[(0.5 - N[(f * N[(Pi * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.2%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around 0 96.2%

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

6. Taylor expanded in f around 0 96.2%

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

   1. *-un-lft-identity 96.3%

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

   2. distribute-rgt-out 96.3%

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

   3. metadata-eval 96.3%

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

8. Applied egg-rr 96.2%

   \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \color{blue}{\left(1 \cdot \left(\pi \cdot -0.041666666666666664\right)\right)}\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]
9. Step-by-step derivation

   1. *-lft-identity 96.3%

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

10. Simplified 96.2%

    \[\leadsto \log \left(\frac{f \cdot \left(-1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \color{blue}{\left(\pi \cdot -0.041666666666666664\right)}\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right) \cdot \frac{-4}{\pi} \]

11. Final simplification 96.2%

    \[\leadsto \frac{-4}{\pi} \cdot \log \left(\frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)}{f} + \frac{2 \cdot \frac{1}{\pi} + f \cdot \left(0.5 - f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)}{f}\right) \]
12. Add Preprocessing

Alternative 5: 96.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \frac{1}{\pi}\\ \frac{-4 \cdot \log \left(\frac{t\_0 - f \cdot \left(0.5 + f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)}{f} + \frac{t\_0 + f \cdot \left(0.5 - f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)}{f}\right)}{\pi} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* 2.0 (/ 1.0 PI))))
   (/
    (*
     -4.0
     (log
      (+
       (/
        (-
         t_0
         (* f (+ 0.5 (* f (+ (* PI -0.125) (* PI 0.08333333333333333))))))
        f)
       (/ (+ t_0 (* f (- 0.5 (* f (* PI -0.041666666666666664))))) f))))
    PI)))

C

double code(double f) {
	double t_0 = 2.0 * (1.0 / ((double) M_PI));
	return (-4.0 * log((((t_0 - (f * (0.5 + (f * ((((double) M_PI) * -0.125) + (((double) M_PI) * 0.08333333333333333)))))) / f) + ((t_0 + (f * (0.5 - (f * (((double) M_PI) * -0.041666666666666664))))) / f)))) / ((double) M_PI);
}

Java

public static double code(double f) {
	double t_0 = 2.0 * (1.0 / Math.PI);
	return (-4.0 * Math.log((((t_0 - (f * (0.5 + (f * ((Math.PI * -0.125) + (Math.PI * 0.08333333333333333)))))) / f) + ((t_0 + (f * (0.5 - (f * (Math.PI * -0.041666666666666664))))) / f)))) / Math.PI;
}

Python

def code(f):
	t_0 = 2.0 * (1.0 / math.pi)
	return (-4.0 * math.log((((t_0 - (f * (0.5 + (f * ((math.pi * -0.125) + (math.pi * 0.08333333333333333)))))) / f) + ((t_0 + (f * (0.5 - (f * (math.pi * -0.041666666666666664))))) / f)))) / math.pi

Julia

function code(f)
	t_0 = Float64(2.0 * Float64(1.0 / pi))
	return Float64(Float64(-4.0 * log(Float64(Float64(Float64(t_0 - Float64(f * Float64(0.5 + Float64(f * Float64(Float64(pi * -0.125) + Float64(pi * 0.08333333333333333)))))) / f) + Float64(Float64(t_0 + Float64(f * Float64(0.5 - Float64(f * Float64(pi * -0.041666666666666664))))) / f)))) / pi)
end

MATLAB

function tmp = code(f)
	t_0 = 2.0 * (1.0 / pi);
	tmp = (-4.0 * log((((t_0 - (f * (0.5 + (f * ((pi * -0.125) + (pi * 0.08333333333333333)))))) / f) + ((t_0 + (f * (0.5 - (f * (pi * -0.041666666666666664))))) / f)))) / pi;
end

Wolfram

code[f_] := Block[{t$95$0 = N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[(-4.0 * N[Log[N[(N[(N[(t$95$0 - N[(f * N[(0.5 + N[(f * N[(N[(Pi * -0.125), $MachinePrecision] + N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision] + N[(N[(t$95$0 + N[(f * N[(0.5 - N[(f * N[(Pi * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]

Derivation

1. Initial program 6.2%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around inf 5.4%

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

   1. associate-*r/ 5.4%

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

7. Simplified 99.0%

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

8. Taylor expanded in f around 0 96.3%

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

   1. *-un-lft-identity 96.3%

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

   2. distribute-rgt-out 96.3%

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

   3. metadata-eval 96.3%

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

10. Applied egg-rr 96.3%

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

    1. *-lft-identity 96.3%

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

12. Simplified 96.3%

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

13. Taylor expanded in f around 0 96.3%

    \[\leadsto \frac{-4 \cdot \log \left(\color{blue}{\frac{f \cdot \left(-1 \cdot \left(f \cdot \left(-0.125 \cdot \pi + 0.08333333333333333 \cdot \pi\right)\right) - 0.5\right) + 2 \cdot \frac{1}{\pi}}{f}} + \frac{f \cdot \left(0.5 + -1 \cdot \left(f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)\right) + 2 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

14. Final simplification 96.3%

    \[\leadsto \frac{-4 \cdot \log \left(\frac{2 \cdot \frac{1}{\pi} - f \cdot \left(0.5 + f \cdot \left(\pi \cdot -0.125 + \pi \cdot 0.08333333333333333\right)\right)}{f} + \frac{2 \cdot \frac{1}{\pi} + f \cdot \left(0.5 - f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)}{f}\right)}{\pi} \]
15. Add Preprocessing

Alternative 6: 96.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \frac{1}{\pi}\\ \frac{-4 \cdot \log \left(\frac{t\_0 + f \cdot \left(0.5 - f \cdot \left(\pi \cdot -0.041666666666666664\right)\right)}{f} + \frac{f \cdot -0.5 + t\_0}{f}\right)}{\pi} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* 2.0 (/ 1.0 PI))))
   (/
    (*
     -4.0
     (log
      (+
       (/ (+ t_0 (* f (- 0.5 (* f (* PI -0.041666666666666664))))) f)
       (/ (+ (* f -0.5) t_0) f))))
    PI)))

C

double code(double f) {
	double t_0 = 2.0 * (1.0 / ((double) M_PI));
	return (-4.0 * log((((t_0 + (f * (0.5 - (f * (((double) M_PI) * -0.041666666666666664))))) / f) + (((f * -0.5) + t_0) / f)))) / ((double) M_PI);
}

Java

public static double code(double f) {
	double t_0 = 2.0 * (1.0 / Math.PI);
	return (-4.0 * Math.log((((t_0 + (f * (0.5 - (f * (Math.PI * -0.041666666666666664))))) / f) + (((f * -0.5) + t_0) / f)))) / Math.PI;
}

Python

def code(f):
	t_0 = 2.0 * (1.0 / math.pi)
	return (-4.0 * math.log((((t_0 + (f * (0.5 - (f * (math.pi * -0.041666666666666664))))) / f) + (((f * -0.5) + t_0) / f)))) / math.pi

Julia

function code(f)
	t_0 = Float64(2.0 * Float64(1.0 / pi))
	return Float64(Float64(-4.0 * log(Float64(Float64(Float64(t_0 + Float64(f * Float64(0.5 - Float64(f * Float64(pi * -0.041666666666666664))))) / f) + Float64(Float64(Float64(f * -0.5) + t_0) / f)))) / pi)
end

MATLAB

function tmp = code(f)
	t_0 = 2.0 * (1.0 / pi);
	tmp = (-4.0 * log((((t_0 + (f * (0.5 - (f * (pi * -0.041666666666666664))))) / f) + (((f * -0.5) + t_0) / f)))) / pi;
end

Wolfram

code[f_] := Block[{t$95$0 = N[(2.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[(-4.0 * N[Log[N[(N[(N[(t$95$0 + N[(f * N[(0.5 - N[(f * N[(Pi * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision] + N[(N[(N[(f * -0.5), $MachinePrecision] + t$95$0), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]]

Derivation

1. Initial program 6.2%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around inf 5.4%

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

   1. associate-*r/ 5.4%

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

7. Simplified 99.0%

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

8. Taylor expanded in f around 0 96.3%

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

   1. *-un-lft-identity 96.3%

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

   2. distribute-rgt-out 96.3%

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

   3. metadata-eval 96.3%

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

10. Applied egg-rr 96.3%

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

    1. *-lft-identity 96.3%

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

12. Simplified 96.3%

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

13. Taylor expanded in f around 0 96.1%

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

14. Final simplification 96.1%

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

Alternative 7: 96.0% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.2%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around 0 96.0%

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

   1. mul-1-neg 96.0%

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

   2. unsub-neg 96.0%

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

7. Simplified 96.0%

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

   1. *-un-lft-identity 96.0%

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

   2. diff-log 96.0%

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

9. Applied egg-rr 96.0%

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

    1. *-lft-identity 96.0%

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

    2. associate-/l/ 96.0%

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

    3. *-commutative 96.0%

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

11. Simplified 96.0%

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

12. Final simplification 96.0%

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

Alternative 8: 4.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{-12}{\pi \cdot {f}^{2}}} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ 1.0 (/ -12.0 (* PI (pow f 2.0)))))

C

double code(double f) {
	return 1.0 / (-12.0 / (((double) M_PI) * pow(f, 2.0)));
}

Java

public static double code(double f) {
	return 1.0 / (-12.0 / (Math.PI * Math.pow(f, 2.0)));
}

Python

def code(f):
	return 1.0 / (-12.0 / (math.pi * math.pow(f, 2.0)))

Julia

function code(f)
	return Float64(1.0 / Float64(-12.0 / Float64(pi * (f ^ 2.0))))
end

MATLAB

function tmp = code(f)
	tmp = 1.0 / (-12.0 / (pi * (f ^ 2.0)));
end

Wolfram

code[f_] := N[(1.0 / N[(-12.0 / N[(Pi * N[Power[f, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.2%

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

3. Simplified 98.9%

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

5. Taylor expanded in f around inf 5.4%

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

   1. associate-*r/ 5.4%

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

7. Simplified 99.0%

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

   1. clear-num 99.0%

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

   2. inv-pow 99.0%

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

   3. +-commutative 99.0%

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

9. Applied egg-rr 99.0%

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

    1. unpow-1 99.0%

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

    2. associate-/r* 99.0%

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

11. Simplified 99.0%

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

12. Taylor expanded in f around 0 96.2%

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

    1. +-commutative 96.2%

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

    2. mul-1-neg 96.2%

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

    3. unsub-neg 96.2%

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

14. Simplified 96.2%

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

15. Taylor expanded in f around inf 4.1%

    \[\leadsto \frac{1}{\color{blue}{\frac{-12}{{f}^{2} \cdot \pi}}} \]

16. Final simplification 4.1%

    \[\leadsto \frac{1}{\frac{-12}{\pi \cdot {f}^{2}}} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(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)))))))))