VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 96.5%

Time: 26.3s

Alternatives: 8

Speedup: 3.3×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{e^{f \cdot \frac{\pi}{-4}} + {\left(e^{\pi}\right)}^{\left(f \cdot 0.25\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {\left(f \cdot \pi\right)}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)}\right) \cdot \frac{-4}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    (+ (exp (* f (/ PI -4.0))) (pow (exp PI) (* f 0.25)))
    (fma
     f
     (* PI 0.5)
     (fma
      (pow f 3.0)
      (* (pow PI 3.0) 0.005208333333333333)
      (* (pow (* f PI) 5.0) 1.6276041666666666e-5)))))
  (/ -4.0 PI)))

C

double code(double f) {
	return log(((exp((f * (((double) M_PI) / -4.0))) + pow(exp(((double) M_PI)), (f * 0.25))) / fma(f, (((double) M_PI) * 0.5), fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), (pow((f * ((double) M_PI)), 5.0) * 1.6276041666666666e-5))))) * (-4.0 / ((double) M_PI));
}

Julia

function code(f)
	return Float64(log(Float64(Float64(exp(Float64(f * Float64(pi / -4.0))) + (exp(pi) ^ Float64(f * 0.25))) / fma(f, Float64(pi * 0.5), fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), Float64((Float64(f * pi) ^ 5.0) * 1.6276041666666666e-5))))) * Float64(-4.0 / pi))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[Exp[N[(f * N[(Pi / -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Power[N[Exp[Pi], $MachinePrecision], N[(f * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(f * N[(Pi * 0.5), $MachinePrecision] + N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(N[Power[N[(f * Pi), $MachinePrecision], 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 7.9%

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

   2. *-commutative 7.9%

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

3. Simplified 7.9%

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

5. Taylor expanded in f around 0 96.7%

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

   1. fma-def 96.7%

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

   2. distribute-rgt-out-- 96.7%

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

   3. metadata-eval 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   4. fma-def 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\mathsf{fma}\left({f}^{3}, 0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   5. distribute-rgt-out-- 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, \color{blue}{{\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   6. metadata-eval 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   7. distribute-rgt-out-- 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \color{blue}{\left({\pi}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} - -8.138020833333333 \cdot 10^{-6}\right)\right)}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   8. metadata-eval 96.7%

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

7. Simplified 96.7%

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

   1. div-inv 96.7%

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

   2. log-prod 96.6%

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

9. Applied egg-rr 96.6%

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

    1. log-rec 96.6%

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

    2. sub-neg 96.6%

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

    3. log-div 96.7%

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

    4. *-commutative 96.7%

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

11. Simplified 96.7%

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

12. Final simplification 96.7%

    \[\leadsto \log \left(\frac{e^{f \cdot \frac{\pi}{-4}} + {\left(e^{\pi}\right)}^{\left(f \cdot 0.25\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {\left(f \cdot \pi\right)}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)}\right) \cdot \frac{-4}{\pi} \]
13. Add Preprocessing

Alternative 2: 96.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(f)
	tmp = (-4.0 * ((log((4.0 / pi)) - log(f)) / pi)) + (-0.08333333333333333 * (pi * (f ^ 2.0)));
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[(-0.08333333333333333 * N[(Pi * N[Power[f, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 7.9%

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

   2. *-commutative 7.9%

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

3. Simplified 7.9%

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

5. Taylor expanded in f around 0 96.7%

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

   1. fma-def 96.7%

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

   2. distribute-rgt-out-- 96.7%

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

   3. metadata-eval 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   4. fma-def 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\mathsf{fma}\left({f}^{3}, 0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   5. distribute-rgt-out-- 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, \color{blue}{{\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   6. metadata-eval 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   7. distribute-rgt-out-- 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \color{blue}{\left({\pi}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} - -8.138020833333333 \cdot 10^{-6}\right)\right)}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   8. metadata-eval 96.7%

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

7. Simplified 96.7%

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

8. Taylor expanded in f around 0 96.4%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]
9. Step-by-step derivation

   1. +-commutative 96.4%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(\left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right) + -1 \cdot \log f\right)}\right) \cdot \frac{-4}{\pi} \]

   2. distribute-lft-out 96.4%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \left(\color{blue}{0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right) + {f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)} + -1 \cdot \log f\right)\right) \cdot \frac{-4}{\pi} \]

   3. fma-def 96.4%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\mathsf{fma}\left(0.5, f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right) + {f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right), -1 \cdot \log f\right)}\right) \cdot \frac{-4}{\pi} \]

10. Simplified 96.4%

    \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \mathsf{fma}\left(0.5, \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right), -\log f\right)\right)} \cdot \frac{-4}{\pi} \]

11. Taylor expanded in f around 0 96.6%

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

12. Final simplification 96.6%

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

Alternative 3: 96.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 7.9%

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

   2. *-commutative 7.9%

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

3. Simplified 7.9%

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

5. Taylor expanded in f around 0 96.7%

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

   1. fma-def 96.7%

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

   2. distribute-rgt-out-- 96.7%

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

   3. metadata-eval 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   4. fma-def 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\mathsf{fma}\left({f}^{3}, 0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   5. distribute-rgt-out-- 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, \color{blue}{{\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   6. metadata-eval 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   7. distribute-rgt-out-- 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \color{blue}{\left({\pi}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} - -8.138020833333333 \cdot 10^{-6}\right)\right)}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   8. metadata-eval 96.7%

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

7. Simplified 96.7%

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

8. Taylor expanded in f around 0 96.4%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]
9. Step-by-step derivation

   1. +-commutative 96.4%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(\left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right) + -1 \cdot \log f\right)}\right) \cdot \frac{-4}{\pi} \]

   2. distribute-lft-out 96.4%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \left(\color{blue}{0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right) + {f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)} + -1 \cdot \log f\right)\right) \cdot \frac{-4}{\pi} \]

   3. fma-def 96.4%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\mathsf{fma}\left(0.5, f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right) + {f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right), -1 \cdot \log f\right)}\right) \cdot \frac{-4}{\pi} \]

10. Simplified 96.4%

    \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \mathsf{fma}\left(0.5, \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right), -\log f\right)\right)} \cdot \frac{-4}{\pi} \]

11. Taylor expanded in f around 0 96.6%

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

    1. div-sub 96.4%

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

    2. sub-neg 96.4%

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

    3. distribute-frac-neg 96.4%

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

    4. log-rec 96.4%

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

    5. *-lft-identity 96.4%

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

    6. metadata-eval 96.4%

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

    7. cancel-sign-sub-inv 96.4%

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

    8. fma-def 96.4%

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

13. Simplified 96.5%

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

14. Final simplification 96.5%

    \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{4}{f \cdot \pi}\right)}{\pi}, {f}^{2} \cdot \left(\pi \cdot -0.08333333333333333\right)\right) \]
15. Add Preprocessing

Alternative 4: 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 7.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 7.9%

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

   2. *-commutative 7.9%

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

3. Simplified 7.9%

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

5. Taylor expanded in f around 0 95.6%

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

   1. associate-/r* 95.6%

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

   2. distribute-rgt-out-- 95.6%

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

   3. metadata-eval 95.6%

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

7. Simplified 95.6%

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

8. Taylor expanded in f around 0 95.8%

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

   1. +-commutative 95.8%

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

   2. mul-1-neg 95.8%

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

   3. log-rec 95.8%

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

   4. log-rec 95.8%

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

   5. mul-1-neg 95.8%

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

   6. +-commutative 95.8%

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

   7. mul-1-neg 95.8%

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

   8. unsub-neg 95.8%

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

10. Simplified 95.8%

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

11. Final simplification 95.8%

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

Alternative 5: 95.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 7.9%

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

   2. *-commutative 7.9%

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

3. Simplified 7.9%

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

5. Taylor expanded in f around 0 95.6%

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

   1. associate-/r* 95.6%

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

   2. distribute-rgt-out-- 95.6%

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

   3. metadata-eval 95.6%

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

7. Simplified 95.6%

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

8. Taylor expanded in f around 0 95.6%

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

   1. *-commutative 95.6%

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

10. Simplified 95.6%

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

11. Final simplification 95.6%

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

Alternative 6: 96.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 7.9%

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

   2. *-commutative 7.9%

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

3. Simplified 7.9%

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

5. Taylor expanded in f around 0 95.6%

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

   1. associate-/r* 95.6%

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

   2. distribute-rgt-out-- 95.6%

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

   3. metadata-eval 95.6%

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

7. Simplified 95.6%

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

8. Taylor expanded in f around 0 95.6%

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

   1. *-commutative 95.6%

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

10. Simplified 95.6%

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

    1. associate-*r/ 95.8%

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

12. Applied egg-rr 95.8%

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

13. Final simplification 95.8%

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

Alternative 7: 4.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 7.9%

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

   2. *-commutative 7.9%

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

3. Simplified 7.9%

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

5. Taylor expanded in f around 0 96.7%

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

   1. fma-def 96.7%

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

   2. distribute-rgt-out-- 96.7%

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

   3. metadata-eval 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   4. fma-def 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\mathsf{fma}\left({f}^{3}, 0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right)}\right) \cdot \frac{-4}{\pi} \]

   5. distribute-rgt-out-- 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, \color{blue}{{\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   6. metadata-eval 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   7. distribute-rgt-out-- 96.7%

      \[\leadsto \log \left(\frac{e^{\frac{\pi}{\frac{-4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \color{blue}{\left({\pi}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} - -8.138020833333333 \cdot 10^{-6}\right)\right)}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

   8. metadata-eval 96.7%

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

7. Simplified 96.7%

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

8. Taylor expanded in f around 0 96.4%

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(-1 \cdot \log f + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]
9. Step-by-step derivation

   1. +-commutative 96.4%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(\left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + 0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)\right) + -1 \cdot \log f\right)}\right) \cdot \frac{-4}{\pi} \]

   2. distribute-lft-out 96.4%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \left(\color{blue}{0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right) + {f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right)\right)} + -1 \cdot \log f\right)\right) \cdot \frac{-4}{\pi} \]

   3. fma-def 96.4%

      \[\leadsto \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\mathsf{fma}\left(0.5, f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right) + {f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\pi \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right)\right), -1 \cdot \log f\right)}\right) \cdot \frac{-4}{\pi} \]

10. Simplified 96.4%

    \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \mathsf{fma}\left(0.5, \mathsf{fma}\left(f, 0, {f}^{2} \cdot \mathsf{fma}\left(0.5, \pi \cdot \left(\pi \cdot 0.08333333333333333\right), 0\right)\right), -\log f\right)\right)} \cdot \frac{-4}{\pi} \]

11. Taylor expanded in f around inf 4.3%

    \[\leadsto \color{blue}{-0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)} \]

12. Final simplification 4.3%

    \[\leadsto -0.08333333333333333 \cdot \left(\pi \cdot {f}^{2}\right) \]
13. Add Preprocessing

Alternative 8: 0.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 7.9%

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

   2. *-commutative 7.9%

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

3. Simplified 7.9%

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

5. Taylor expanded in f around 0 95.6%

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

6. Taylor expanded in f around inf 0.7%

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

   1. distribute-rgt-out 0.7%

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

   2. distribute-rgt-out-- 0.7%

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

   3. metadata-eval 0.7%

      \[\leadsto \log \left(\frac{\pi}{\pi \cdot \color{blue}{0.5}} \cdot \left(-0.25 + 0.25\right)\right) \cdot \frac{-4}{\pi} \]

   4. metadata-eval 0.7%

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

   5. mul0-rgt 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 2024010 
(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)))))))))