VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.0% → 96.6%

Time: 29.1s

Alternatives: 7

Speedup: 3.3×

• could not determine a ground truth

  1. f = -1.660512399668033e+246

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 7.0% 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: 96.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ -\frac{\log \left(\frac{2}{\mathsf{fma}\left({\left(\pi \cdot f\right)}^{3}, 0.005208333333333333, \mathsf{fma}\left(1.6276041666666666 \cdot 10^{-5}, {f}^{5} \cdot {\pi}^{5}, \pi \cdot \left(f \cdot 0.5\right)\right)\right)} \cdot \cosh \left(\pi \cdot \left(f \cdot 0.25\right)\right)\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (/
   (log
    (*
     (/
      2.0
      (fma
       (pow (* PI f) 3.0)
       0.005208333333333333
       (fma
        1.6276041666666666e-5
        (* (pow f 5.0) (pow PI 5.0))
        (* PI (* f 0.5)))))
     (cosh (* PI (* f 0.25)))))
   (* PI 0.25))))

C

double code(double f) {
	return -(log(((2.0 / fma(pow((((double) M_PI) * f), 3.0), 0.005208333333333333, fma(1.6276041666666666e-5, (pow(f, 5.0) * pow(((double) M_PI), 5.0)), (((double) M_PI) * (f * 0.5))))) * cosh((((double) M_PI) * (f * 0.25))))) / (((double) M_PI) * 0.25));
}

Julia

function code(f)
	return Float64(-Float64(log(Float64(Float64(2.0 / fma((Float64(pi * f) ^ 3.0), 0.005208333333333333, fma(1.6276041666666666e-5, Float64((f ^ 5.0) * (pi ^ 5.0)), Float64(pi * Float64(f * 0.5))))) * cosh(Float64(pi * Float64(f * 0.25))))) / Float64(pi * 0.25)))
end

Wolfram

code[f_] := (-N[(N[Log[N[(N[(2.0 / N[(N[Power[N[(Pi * f), $MachinePrecision], 3.0], $MachinePrecision] * 0.005208333333333333 + N[(1.6276041666666666e-5 * N[(N[Power[f, 5.0], $MachinePrecision] * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision] + N[(Pi * N[(f * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cosh[N[(Pi * N[(f * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 7.2%

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

2. Taylor expanded in f around 0 96.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. associate-+r+ 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}}\right) \]

   2. +-commutative 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right)}}\right) \]

   3. *-commutative 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) \cdot {f}^{3}} + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right)}\right) \]

   4. distribute-rgt-out-- 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)} \cdot {f}^{3} + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right)}\right) \]

   5. associate-*l* 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{\pi}^{3} \cdot \left(\left(0.0026041666666666665 - -0.0026041666666666665\right) \cdot {f}^{3}\right)} + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right)}\right) \]

   6. fma-def 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({\pi}^{3}, \left(0.0026041666666666665 - -0.0026041666666666665\right) \cdot {f}^{3}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right)}}\right) \]

   7. metadata-eval 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({\pi}^{3}, \color{blue}{0.005208333333333333} \cdot {f}^{3}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right)}\right) \]

4. Simplified 96.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)}}\right) \]
5. Step-by-step derivation

   1. log-div 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}\right) - \log \left(\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)\right)\right)} \]

   2. cosh-undef 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \color{blue}{\left(2 \cdot \cosh \left(\frac{\pi}{4} \cdot f\right)\right)} - \log \left(\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)\right)\right) \]

   3. div-inv 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(2 \cdot \cosh \left(\color{blue}{\left(\pi \cdot \frac{1}{4}\right)} \cdot f\right)\right) - \log \left(\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)\right)\right) \]

   4. metadata-eval 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(2 \cdot \cosh \left(\left(\pi \cdot \color{blue}{0.25}\right) \cdot f\right)\right) - \log \left(\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)\right)\right) \]

6. Applied egg-rr 96.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)\right) - \log \left(\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. associate-*l/ 96.3%

      \[\leadsto -\color{blue}{\frac{1 \cdot \left(\log \left(2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)\right) - \log \left(\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)\right)\right)}{\frac{\pi}{4}}} \]

8. Applied egg-rr 96.3%

   \[\leadsto -\color{blue}{\frac{\log \left(\frac{2 \cdot \cosh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)}\right)}{\pi \cdot 0.25}} \]
9. Step-by-step derivation

10. Simplified 96.3%

    \[\leadsto -\color{blue}{\frac{\log \left(\frac{2}{\mathsf{fma}\left({\left(\pi \cdot f\right)}^{3}, 0.005208333333333333, \mathsf{fma}\left(1.6276041666666666 \cdot 10^{-5}, {f}^{5} \cdot {\pi}^{5}, \pi \cdot \left(0.5 \cdot f\right)\right)\right)} \cdot \cosh \left(\pi \cdot \left(f \cdot 0.25\right)\right)\right)}{\pi \cdot 0.25}} \]

11. Final simplification 96.3%

    \[\leadsto -\frac{\log \left(\frac{2}{\mathsf{fma}\left({\left(\pi \cdot f\right)}^{3}, 0.005208333333333333, \mathsf{fma}\left(1.6276041666666666 \cdot 10^{-5}, {f}^{5} \cdot {\pi}^{5}, \pi \cdot \left(f \cdot 0.5\right)\right)\right)} \cdot \cosh \left(\pi \cdot \left(f \cdot 0.25\right)\right)\right)}{\pi \cdot 0.25} \]

Alternative 2: 96.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{e^{f \cdot \frac{\pi}{4}} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\mathsf{fma}\left(\pi \cdot 0.5, f, {\pi}^{3} \cdot \left(0.005208333333333333 \cdot {f}^{3}\right)\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    (+ (exp (* f (/ PI 4.0))) (exp (* (/ PI 4.0) (- f))))
    (fma (* PI 0.5) f (* (pow PI 3.0) (* 0.005208333333333333 (pow f 3.0))))))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(((exp((f * (((double) M_PI) / 4.0))) + exp(((((double) M_PI) / 4.0) * -f))) / fma((((double) M_PI) * 0.5), f, (pow(((double) M_PI), 3.0) * (0.005208333333333333 * pow(f, 3.0)))))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(Float64(Float64(exp(Float64(f * Float64(pi / 4.0))) + exp(Float64(Float64(pi / 4.0) * Float64(-f)))) / fma(Float64(pi * 0.5), f, Float64((pi ^ 3.0) * Float64(0.005208333333333333 * (f ^ 3.0)))))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[Exp[N[(f * N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * (-f)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(Pi * 0.5), $MachinePrecision] * f + N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(0.005208333333333333 * N[Power[f, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.2%

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

2. Taylor expanded in f around 0 95.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}}\right) \]
3. Step-by-step derivation

   1. fma-def 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(0.25 \cdot \pi - -0.25 \cdot \pi, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}}\right) \]

   2. distribute-rgt-out-- 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \]

   3. metadata-eval 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(\pi \cdot \color{blue}{0.5}, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right) \]

   4. *-commutative 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(\pi \cdot 0.5, f, \color{blue}{\left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) \cdot {f}^{3}}\right)}\right) \]

   5. distribute-rgt-out-- 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(\pi \cdot 0.5, f, \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)} \cdot {f}^{3}\right)}\right) \]

   6. associate-*l* 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(\pi \cdot 0.5, f, \color{blue}{{\pi}^{3} \cdot \left(\left(0.0026041666666666665 - -0.0026041666666666665\right) \cdot {f}^{3}\right)}\right)}\right) \]

   7. metadata-eval 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(\pi \cdot 0.5, f, {\pi}^{3} \cdot \left(\color{blue}{0.005208333333333333} \cdot {f}^{3}\right)\right)}\right) \]

4. Simplified 95.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(\pi \cdot 0.5, f, {\pi}^{3} \cdot \left(0.005208333333333333 \cdot {f}^{3}\right)\right)}}\right) \]

5. Final simplification 95.9%

   \[\leadsto \log \left(\frac{e^{f \cdot \frac{\pi}{4}} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\mathsf{fma}\left(\pi \cdot 0.5, f, {\pi}^{3} \cdot \left(0.005208333333333333 \cdot {f}^{3}\right)\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}} \]

Alternative 3: 96.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double f) {
	return log(fma(f, (((double) M_PI) * 0.08333333333333333), (2.0 / (((double) M_PI) * (f * 0.5))))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(fma(f, Float64(pi * 0.08333333333333333), Float64(2.0 / Float64(pi * Float64(f * 0.5))))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

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

Derivation

1. Initial program 7.2%

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

2. Taylor expanded in f around 0 95.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(2 \cdot \frac{1}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f} + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right)\right)\right)\right)} \]

4. Simplified 95.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5} \cdot 1\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)} \]
5. Step-by-step derivation

   1. fma-udef 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right) \cdot -2 + \frac{0.0625 \cdot \pi}{0.5} \cdot 1}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   2. pow-div 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\color{blue}{{\pi}^{\left(3 - 2\right)}} \cdot 0.020833333333333332\right) \cdot -2 + \frac{0.0625 \cdot \pi}{0.5} \cdot 1, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   3. metadata-eval 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left({\pi}^{\color{blue}{1}} \cdot 0.020833333333333332\right) \cdot -2 + \frac{0.0625 \cdot \pi}{0.5} \cdot 1, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   4. pow1 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\color{blue}{\pi} \cdot 0.020833333333333332\right) \cdot -2 + \frac{0.0625 \cdot \pi}{0.5} \cdot 1, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   5. *-rgt-identity 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \color{blue}{\frac{0.0625 \cdot \pi}{0.5}}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   6. div-inv 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \color{blue}{\left(0.0625 \cdot \pi\right) \cdot \frac{1}{0.5}}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   7. *-commutative 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \color{blue}{\left(\pi \cdot 0.0625\right)} \cdot \frac{1}{0.5}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   8. metadata-eval 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \left(\pi \cdot 0.0625\right) \cdot \color{blue}{2}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

6. Applied egg-rr 95.9%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\left(\pi \cdot 0.020833333333333332\right) \cdot -2 + \left(\pi \cdot 0.0625\right) \cdot 2}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]
7. Step-by-step derivation

   1. associate-*l* 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.020833333333333332 \cdot -2\right)} + \left(\pi \cdot 0.0625\right) \cdot 2, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   2. metadata-eval 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot \color{blue}{-0.041666666666666664} + \left(\pi \cdot 0.0625\right) \cdot 2, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   3. metadata-eval 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot \color{blue}{\left(-0.041666666666666664\right)} + \left(\pi \cdot 0.0625\right) \cdot 2, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   4. associate-*l* 95.9%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot \left(-0.041666666666666664\right) + \color{blue}{\pi \cdot \left(0.0625 \cdot 2\right)}, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \]

   5. metadata-eval 95.9%

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

   6. distribute-lft-out 95.9%

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

   7. metadata-eval 95.9%

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

   8. metadata-eval 95.9%

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

8. Simplified 95.9%

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

9. Final simplification 95.9%

   \[\leadsto \log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \]

Alternative 4: 95.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

2. Taylor expanded in f around 0 96.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. associate-+r+ 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}}\right) \]

   2. +-commutative 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right)}}\right) \]

   3. *-commutative 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) \cdot {f}^{3}} + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right)}\right) \]

   4. distribute-rgt-out-- 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)} \cdot {f}^{3} + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right)}\right) \]

   5. associate-*l* 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{\pi}^{3} \cdot \left(\left(0.0026041666666666665 - -0.0026041666666666665\right) \cdot {f}^{3}\right)} + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right)}\right) \]

   6. fma-def 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({\pi}^{3}, \left(0.0026041666666666665 - -0.0026041666666666665\right) \cdot {f}^{3}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right)}}\right) \]

   7. metadata-eval 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({\pi}^{3}, \color{blue}{0.005208333333333333} \cdot {f}^{3}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right)}\right) \]

4. Simplified 96.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)}}\right) \]

5. Taylor expanded in f around 0 95.2%

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

   1. neg-mul-1 95.2%

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

   2. log-rec 95.2%

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

   3. +-commutative 95.2%

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

   4. log-rec 95.2%

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

   5. sub-neg 95.2%

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

7. Simplified 95.2%

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

8. Final simplification 95.2%

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

Alternative 5: 95.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

2. Taylor expanded in f around 0 95.1%

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

   1. distribute-rgt-out-- 95.1%

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

   2. metadata-eval 95.1%

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

4. Simplified 95.1%

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

   1. associate-/r* 95.1%

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

   2. *-commutative 95.1%

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

   3. associate-/l/ 95.1%

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

   4. diff-log 95.2%

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

   5. associate-*l/ 95.2%

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

   6. *-un-lft-identity 95.2%

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

   7. diff-log 95.2%

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

   8. associate-/l/ 95.2%

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

   9. *-commutative 95.2%

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

   10. associate-/r* 95.2%

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

   11. associate-*l* 95.2%

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

   12. *-commutative 95.2%

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

   13. div-inv 95.2%

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

   14. metadata-eval 95.2%

       \[\leadsto -\frac{\log \left(\frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)}{\pi \cdot \color{blue}{0.25}} \]

6. Applied egg-rr 95.2%

   \[\leadsto -\color{blue}{\frac{\log \left(\frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)}{\pi \cdot 0.25}} \]
7. Step-by-step derivation

   1. *-lft-identity 95.2%

      \[\leadsto -\frac{\color{blue}{1 \cdot \log \left(\frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)}}{\pi \cdot 0.25} \]

   2. *-commutative 95.2%

      \[\leadsto -\frac{1 \cdot \log \left(\frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)}{\color{blue}{0.25 \cdot \pi}} \]

   3. times-frac 95.2%

      \[\leadsto -\color{blue}{\frac{1}{0.25} \cdot \frac{\log \left(\frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)}{\pi}} \]

   4. metadata-eval 95.2%

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

   5. associate-*r* 95.2%

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

   6. *-commutative 95.2%

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

   7. *-commutative 95.2%

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

   8. associate-/r* 95.2%

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

   9. metadata-eval 95.2%

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

8. Simplified 95.2%

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

9. Final simplification 95.2%

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

Alternative 6: 95.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

2. Taylor expanded in f around 0 96.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. associate-+r+ 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}}\right) \]

   2. +-commutative 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right)}}\right) \]

   3. *-commutative 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) \cdot {f}^{3}} + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right)}\right) \]

   4. distribute-rgt-out-- 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)} \cdot {f}^{3} + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right)}\right) \]

   5. associate-*l* 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{{\pi}^{3} \cdot \left(\left(0.0026041666666666665 - -0.0026041666666666665\right) \cdot {f}^{3}\right)} + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right)}\right) \]

   6. fma-def 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({\pi}^{3}, \left(0.0026041666666666665 - -0.0026041666666666665\right) \cdot {f}^{3}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right)}}\right) \]

   7. metadata-eval 96.2%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({\pi}^{3}, \color{blue}{0.005208333333333333} \cdot {f}^{3}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f\right)}\right) \]

4. Simplified 96.2%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)}}\right) \]
5. Step-by-step derivation

   1. associate-*l/ 96.3%

      \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)}\right)}{\frac{\pi}{4}}} \]

6. Applied egg-rr 96.3%

   \[\leadsto -\color{blue}{\frac{\log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)}\right)}{\pi \cdot 0.25}} \]
7. Step-by-step derivation

   1. associate-/l* 96.3%

      \[\leadsto -\frac{\log \color{blue}{\left(\frac{2}{\frac{\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)}{\cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}}\right)}}{\pi \cdot 0.25} \]

   2. associate-/r/ 96.3%

      \[\leadsto -\frac{\log \color{blue}{\left(\frac{2}{\mathsf{fma}\left({\pi}^{3}, 0.005208333333333333 \cdot {f}^{3}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)} \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)\right)}}{\pi \cdot 0.25} \]

   3. *-commutative 96.3%

      \[\leadsto -\frac{\log \left(\frac{2}{\mathsf{fma}\left({\pi}^{3}, \color{blue}{{f}^{3} \cdot 0.005208333333333333}, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)} \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)\right)}{\pi \cdot 0.25} \]

   4. associate-*l* 96.3%

      \[\leadsto -\frac{\log \left(\frac{2}{\mathsf{fma}\left({\pi}^{3}, {f}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)} \cdot \cosh \color{blue}{\left(\pi \cdot \left(0.25 \cdot f\right)\right)}\right)}{\pi \cdot 0.25} \]

   5. *-commutative 96.3%

      \[\leadsto -\frac{\log \left(\frac{2}{\mathsf{fma}\left({\pi}^{3}, {f}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)} \cdot \cosh \left(\pi \cdot \color{blue}{\left(f \cdot 0.25\right)}\right)\right)}{\pi \cdot 0.25} \]

8. Simplified 96.3%

   \[\leadsto -\color{blue}{\frac{\log \left(\frac{2}{\mathsf{fma}\left({\pi}^{3}, {f}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left(1.6276041666666666 \cdot 10^{-5} \cdot {f}^{5}\right)\right)\right)} \cdot \cosh \left(\pi \cdot \left(f \cdot 0.25\right)\right)\right)}{\pi \cdot 0.25}} \]

9. Taylor expanded in f around 0 95.2%

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

    1. mul-1-neg 95.2%

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

    2. log-rec 95.2%

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

    3. +-commutative 95.2%

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

    4. log-rec 95.2%

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

    5. sub-neg 95.2%

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

    6. metadata-eval 95.2%

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

    7. associate-*l/ 95.2%

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

    8. log-div 95.2%

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

    9. associate-*r/ 95.2%

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

    10. *-commutative 95.2%

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

11. Simplified 95.2%

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

12. Final simplification 95.2%

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

Alternative 7: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{-\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* 4.0 (/ (- (log 7.62939453125e-6)) PI)))

C

double code(double f) {
	return 4.0 * (-log(7.62939453125e-6) / ((double) M_PI));
}

Java

public static double code(double f) {
	return 4.0 * (-Math.log(7.62939453125e-6) / Math.PI);
}

Python

def code(f):
	return 4.0 * (-math.log(7.62939453125e-6) / math.pi)

Julia

function code(f)
	return Float64(4.0 * Float64(Float64(-log(7.62939453125e-6)) / pi))
end

MATLAB

function tmp = code(f)
	tmp = 4.0 * (-log(7.62939453125e-6) / pi);
end

Wolfram

code[f_] := N[(4.0 * N[((-N[Log[7.62939453125e-6], $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. Applied egg-rr 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{262144}}\right) \]

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}} \]

5. Final simplification 1.6%

   \[\leadsto 4 \cdot \frac{-\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi} \]

Reproduce

herbie shell --seed 2023182 
(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)))))))))