VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 98.8%

Time: 17.1s

Alternatives: 10

Speedup: 4.9×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double f) {
	return -4.0 * (1.0 / (((double) M_PI) / log1p(((1.0 / expm1((f * (pow(sqrt(((double) M_PI)), 2.0) * 0.5)))) + (-1.0 + (-1.0 / expm1((f * (((double) M_PI) * -0.5)))))))));
}

Java

public static double code(double f) {
	return -4.0 * (1.0 / (Math.PI / Math.log1p(((1.0 / Math.expm1((f * (Math.pow(Math.sqrt(Math.PI), 2.0) * 0.5)))) + (-1.0 + (-1.0 / Math.expm1((f * (Math.PI * -0.5)))))))));
}

Python

def code(f):
	return -4.0 * (1.0 / (math.pi / math.log1p(((1.0 / math.expm1((f * (math.pow(math.sqrt(math.pi), 2.0) * 0.5)))) + (-1.0 + (-1.0 / math.expm1((f * (math.pi * -0.5)))))))))

Julia

function code(f)
	return Float64(-4.0 * Float64(1.0 / Float64(pi / log1p(Float64(Float64(1.0 / expm1(Float64(f * Float64((sqrt(pi) ^ 2.0) * 0.5)))) + Float64(-1.0 + Float64(-1.0 / expm1(Float64(f * Float64(pi * -0.5))))))))))
end

Wolfram

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

Derivation

1. Initial program 6.4%

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

3. Simplified 98.8%

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

5. Taylor expanded in f around inf 6.3%

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

7. Simplified 98.9%

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

   1. clear-num 98.9%

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

   2. inv-pow 98.9%

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

9. Applied egg-rr 98.9%

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

    1. unpow-1 98.9%

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

    2. associate-*l* 98.9%

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

11. Simplified 98.9%

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

    1. log1p-expm1-u 98.9%

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

    2. expm1-undefine 98.9%

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

    3. add-exp-log 98.9%

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

13. Applied egg-rr 98.9%

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

    1. sub-neg 98.9%

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

    2. +-commutative 98.9%

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

    3. metadata-eval 98.9%

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

    4. distribute-neg-frac 98.9%

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

    5. sub-neg 98.9%

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

    6. metadata-eval 98.9%

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

    7. associate-+l- 98.9%

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

15. Simplified 98.9%

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

    1. add-sqr-sqrt 98.9%

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

    2. pow2 98.9%

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

17. Applied egg-rr 98.9%

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

18. Final simplification 98.9%

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

Alternative 2: 98.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

3. Simplified 98.8%

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

5. Taylor expanded in f around inf 6.3%

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

7. Simplified 98.9%

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

   1. clear-num 98.9%

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

   2. inv-pow 98.9%

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

9. Applied egg-rr 98.9%

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

    1. unpow-1 98.9%

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

    2. associate-*l* 98.9%

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

11. Simplified 98.9%

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

    1. log1p-expm1-u 98.9%

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

    2. expm1-undefine 98.9%

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

    3. add-exp-log 98.9%

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

13. Applied egg-rr 98.9%

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

    1. sub-neg 98.9%

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

    2. +-commutative 98.9%

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

    3. metadata-eval 98.9%

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

    4. distribute-neg-frac 98.9%

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

    5. sub-neg 98.9%

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

    6. metadata-eval 98.9%

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

    7. associate-+l- 98.9%

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

15. Simplified 98.9%

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

    1. un-div-inv 98.9%

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

    2. sub-neg 98.9%

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

    3. metadata-eval 98.9%

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

17. Applied egg-rr 98.9%

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

18. Final simplification 98.9%

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

Alternative 3: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Derivation

1. Initial program 6.4%

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

3. Simplified 98.8%

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

5. Taylor expanded in f around inf 6.3%

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

7. Simplified 98.9%

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

Alternative 4: 96.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(f)
	tmp = (-4.0 * ((log((4.0 / pi)) - log(f)) / pi)) - ((f ^ 2.0) * (pi * 0.08333333333333333));
end

Wolfram

code[f_] := N[(N[(-4.0 * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] - N[(N[Power[f, 2.0], $MachinePrecision] * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.4%

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

3. Simplified 98.8%

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

5. Taylor expanded in f around 0 96.1%

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

   1. mul-1-neg 96.1%

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

   2. unsub-neg 96.1%

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

   3. mul-1-neg 96.1%

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

   4. unsub-neg 96.1%

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

   5. distribute-rgt-out 96.1%

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

   6. metadata-eval 96.1%

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

7. Simplified 96.1%

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

Alternative 5: 96.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

double code(double f) {
	return -4.0 * (log((((pow(f, 2.0) * (((double) M_PI) * 0.08333333333333333)) + (4.0 * (1.0 / ((double) M_PI)))) / f)) / ((double) M_PI));
}

Java

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

Python

def code(f):
	return -4.0 * (math.log((((math.pow(f, 2.0) * (math.pi * 0.08333333333333333)) + (4.0 * (1.0 / math.pi))) / f)) / math.pi)

Julia

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

MATLAB

function tmp = code(f)
	tmp = -4.0 * (log(((((f ^ 2.0) * (pi * 0.08333333333333333)) + (4.0 * (1.0 / pi))) / f)) / pi);
end

Wolfram

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

Derivation

1. Initial program 6.4%

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

3. Simplified 98.8%

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

5. Taylor expanded in f around inf 6.3%

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

7. Simplified 98.9%

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

8. Taylor expanded in f around 0 96.0%

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

   1. associate--l+ 96.0%

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

   2. *-commutative 96.0%

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

   3. *-commutative 96.0%

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

   4. distribute-rgt-out 96.0%

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

   5. metadata-eval 96.0%

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

10. Applied egg-rr 96.0%

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

    1. distribute-lft-out-- 96.0%

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

    2. distribute-lft-out 96.0%

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

    3. metadata-eval 96.0%

       \[\leadsto -4 \cdot \frac{\log \left(\frac{{f}^{2} \cdot \left(\pi \cdot \left(-0.08333333333333333 + \color{blue}{0.16666666666666666}\right)\right) + 4 \cdot \frac{1}{\pi}}{f}\right)}{\pi} \]

    4. metadata-eval 96.0%

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

12. Simplified 96.0%

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

Alternative 6: 96.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

3. Simplified 98.8%

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

5. Taylor expanded in f around 0 95.6%

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

   1. mul-1-neg 95.6%

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

   2. unsub-neg 95.6%

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

7. Simplified 95.6%

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

Alternative 7: 96.0% 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 6.4%

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

3. Simplified 98.8%

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

5. Taylor expanded in f around 0 95.6%

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

   1. mul-1-neg 95.6%

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

   2. unsub-neg 95.6%

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

7. Simplified 95.6%

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

   1. associate-*r/ 95.6%

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

   2. diff-log 95.6%

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

9. Applied egg-rr 95.6%

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

Alternative 8: 95.8% accurate, 4.9× speedup

Math

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

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

Derivation

1. Initial program 6.4%

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

3. Simplified 98.8%

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

5. Taylor expanded in f around 0 95.4%

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

   1. *-commutative 95.4%

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

7. Simplified 95.4%

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

Alternative 9: 1.6% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(f)
	tmp = (-4.0 / pi) * log((0.08333333333333333 * (pi * f)));
end

Wolfram

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

Derivation

1. Initial program 6.4%

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

3. Simplified 98.8%

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

5. Taylor expanded in f around 0 95.9%

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

6. Taylor expanded in f around inf 1.6%

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

   1. distribute-rgt-out 1.6%

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

   2. metadata-eval 1.6%

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

   3. distribute-rgt-out 1.6%

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

   4. metadata-eval 1.6%

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

   5. distribute-lft-out-- 1.6%

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

   6. metadata-eval 1.6%

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

8. Simplified 1.6%

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

9. Taylor expanded in f around 0 1.6%

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

10. Final simplification 1.6%

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

Alternative 10: 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 6.4%

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

3. Simplified 98.8%

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

6. Applied egg-rr 0.7%

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

   1. +-inverses 0.7%

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

8. Simplified 0.7%

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

9. Final simplification 0.7%

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

Reproduce

herbie shell --seed 2024137 
(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)))))))))