VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 96.4%

Time: 16.4s

Alternatives: 6

Speedup: 4.6×

• could not determine a ground truth

  1. f = 1343555299.343332

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: 6.8% 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: 96.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in f around inf

   \[\leadsto \mathsf{neg}\left(\color{blue}{4 \cdot \frac{\log \left(\frac{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)}}\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \frac{\color{blue}{\frac{2}{\frac{1}{2}}}}{\mathsf{PI}\left(\right)}\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \frac{\frac{2}{\color{blue}{\frac{1}{4} - \frac{-1}{4}}}}{\mathsf{PI}\left(\right)}\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \color{blue}{\frac{2}{\left(\frac{1}{4} - \frac{-1}{4}\right) \cdot \mathsf{PI}\left(\right)}}\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \frac{2}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)}}\right) \]

   8. distribute-rgt-out-- N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \frac{2}{\color{blue}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \frac{\color{blue}{2 \cdot 1}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \color{blue}{\left(2 \cdot \frac{1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}\right)}\right) \]

5. Simplified 7.5%

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

6. Taylor expanded in f around 0

   \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} + e^{f \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right)}}{\color{blue}{f \cdot \left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) + {f}^{2} \cdot \left(\left(\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} + {f}^{2} \cdot \left(\frac{1}{122880} \cdot {\mathsf{PI}\left(\right)}^{5} - \frac{-1}{122880} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right) - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot \frac{4}{\mathsf{PI}\left(\right)}\right) \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} + e^{f \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right)}}{\color{blue}{f \cdot \left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) + {f}^{2} \cdot \left(\left(\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} + {f}^{2} \cdot \left(\frac{1}{122880} \cdot {\mathsf{PI}\left(\right)}^{5} - \frac{-1}{122880} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right) - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot \frac{4}{\mathsf{PI}\left(\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} + e^{f \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right)}}{f \cdot \left(\color{blue}{\left({f}^{2} \cdot \left(\left(\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} + {f}^{2} \cdot \left(\frac{1}{122880} \cdot {\mathsf{PI}\left(\right)}^{5} - \frac{-1}{122880} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right) - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \frac{4}{\mathsf{PI}\left(\right)}\right) \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} + e^{f \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right)}}{f \cdot \color{blue}{\left({f}^{2} \cdot \left(\left(\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} + {f}^{2} \cdot \left(\frac{1}{122880} \cdot {\mathsf{PI}\left(\right)}^{5} - \frac{-1}{122880} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right) - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \frac{4}{\mathsf{PI}\left(\right)}\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} + e^{f \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right)}}{f \cdot \color{blue}{\mathsf{fma}\left({f}^{2}, \left(\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} + {f}^{2} \cdot \left(\frac{1}{122880} \cdot {\mathsf{PI}\left(\right)}^{5} - \frac{-1}{122880} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right) - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot \frac{4}{\mathsf{PI}\left(\right)}\right) \]

8. Simplified 97.2%

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

9. Final simplification 97.2%

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

Alternative 2: 96.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ -\mathsf{fma}\left(4, \frac{\left(-\log f\right) - \log \left(0.25 \cdot \pi\right)}{\pi}, \left(f \cdot f\right) \cdot \left(\pi \cdot \mathsf{fma}\left(\pi \cdot \pi, \left(f \cdot f\right) \cdot -0.0008680555555555555, 0.08333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (fma
   4.0
   (/ (- (- (log f)) (log (* 0.25 PI))) PI)
   (*
    (* f f)
    (*
     PI
     (fma
      (* PI PI)
      (* (* f f) -0.0008680555555555555)
      0.08333333333333333))))))

C

double code(double f) {
	return -fma(4.0, ((-log(f) - log((0.25 * ((double) M_PI)))) / ((double) M_PI)), ((f * f) * (((double) M_PI) * fma((((double) M_PI) * ((double) M_PI)), ((f * f) * -0.0008680555555555555), 0.08333333333333333))));
}

Julia

function code(f)
	return Float64(-fma(4.0, Float64(Float64(Float64(-log(f)) - log(Float64(0.25 * pi))) / pi), Float64(Float64(f * f) * Float64(pi * fma(Float64(pi * pi), Float64(Float64(f * f) * -0.0008680555555555555), 0.08333333333333333)))))
end

Wolfram

code[f_] := (-N[(4.0 * N[(N[((-N[Log[f], $MachinePrecision]) - N[Log[N[(0.25 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] + N[(N[(f * f), $MachinePrecision] * N[(Pi * N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(f * f), $MachinePrecision] * -0.0008680555555555555), $MachinePrecision] + 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in f around 0

   \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{f \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + f \cdot \left(\frac{1}{16} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}}{{\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)\right) + 2 \cdot \frac{1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{f}\right)}\right) \]

5. Simplified 97.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(f, f \cdot \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \pi \cdot 0.125\right), \frac{4}{\pi}\right)}{f}\right)} \]

6. Taylor expanded in f around 0

   \[\leadsto \mathsf{neg}\left(\color{blue}{\left(4 \cdot \frac{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f}{\mathsf{PI}\left(\right)} + {f}^{2} \cdot \left(\frac{-1}{8} \cdot \left({f}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right) + \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{8} \cdot \left({f}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right) + \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

   2. lower-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \color{blue}{\frac{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f}{\mathsf{PI}\left(\right)}}, {f}^{2} \cdot \left(\frac{-1}{8} \cdot \left({f}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right) + \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + \color{blue}{\left(\mathsf{neg}\left(\log f\right)\right)}}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{8} \cdot \left({f}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right) + \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   4. unsub-neg N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\color{blue}{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \log f}}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{8} \cdot \left({f}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right) + \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\color{blue}{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \log f}}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{8} \cdot \left({f}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right) + \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   6. lower-log.f64 N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\color{blue}{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right)} - \log f}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{8} \cdot \left({f}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right) + \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. lower-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\log \color{blue}{\left(\frac{4}{\mathsf{PI}\left(\right)}\right)} - \log f}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{8} \cdot \left({f}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right) + \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   8. lower-PI.f64 N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\color{blue}{\mathsf{PI}\left(\right)}}\right) - \log f}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{8} \cdot \left({f}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right) + \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   9. lower-log.f64 N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \color{blue}{\log f}}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{8} \cdot \left({f}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right) + \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   10. lower-PI.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \log f}{\color{blue}{\mathsf{PI}\left(\right)}}, {f}^{2} \cdot \left(\frac{-1}{8} \cdot \left({f}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right) + \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \log f}{\mathsf{PI}\left(\right)}, \color{blue}{{f}^{2} \cdot \left(\frac{-1}{8} \cdot \left({f}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right) + \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

8. Simplified 97.1%

   \[\leadsto -\color{blue}{\mathsf{fma}\left(4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, \left(f \cdot f\right) \cdot \left(\pi \cdot \mathsf{fma}\left(\left(f \cdot f\right) \cdot -0.125, \left(\pi \cdot \pi\right) \cdot 0.006944444444444444, 0.08333333333333333\right)\right)\right)} \]

9. Taylor expanded in f around 0

   \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \color{blue}{\frac{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \log f}{\mathsf{PI}\left(\right)}}, \left(f \cdot f\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(f \cdot f\right) \cdot \frac{-1}{8}, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{144}, \frac{1}{12}\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \color{blue}{\frac{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \log f}{\mathsf{PI}\left(\right)}}, \left(f \cdot f\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(f \cdot f\right) \cdot \frac{-1}{8}, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{144}, \frac{1}{12}\right)\right)\right)\right) \]

11. Simplified 97.1%

    \[\leadsto -\mathsf{fma}\left(4, \color{blue}{\frac{\left(-\log \left(\pi \cdot 0.25\right)\right) - \log f}{\pi}}, \left(f \cdot f\right) \cdot \left(\pi \cdot \mathsf{fma}\left(\left(f \cdot f\right) \cdot -0.125, \left(\pi \cdot \pi\right) \cdot 0.006944444444444444, 0.08333333333333333\right)\right)\right) \]

12. Taylor expanded in f around 0

    \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\left(\mathsf{neg}\left(\log \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)\right) - \log f}{\mathsf{PI}\left(\right)}, \left(f \cdot f\right) \cdot \color{blue}{\left(\frac{-1}{1152} \cdot \left({f}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{12} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]
13. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\left(\mathsf{neg}\left(\log \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)\right) - \log f}{\mathsf{PI}\left(\right)}, \left(f \cdot f\right) \cdot \left(\color{blue}{\left(\frac{-1}{1152} \cdot {f}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + \frac{1}{12} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    2. unpow3 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\left(\mathsf{neg}\left(\log \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)\right) - \log f}{\mathsf{PI}\left(\right)}, \left(f \cdot f\right) \cdot \left(\left(\frac{-1}{1152} \cdot {f}^{2}\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)} + \frac{1}{12} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\left(\mathsf{neg}\left(\log \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)\right) - \log f}{\mathsf{PI}\left(\right)}, \left(f \cdot f\right) \cdot \left(\left(\frac{-1}{1152} \cdot {f}^{2}\right) \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{12} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    4. associate-*r* N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\left(\mathsf{neg}\left(\log \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)\right) - \log f}{\mathsf{PI}\left(\right)}, \left(f \cdot f\right) \cdot \left(\color{blue}{\left(\left(\frac{-1}{1152} \cdot {f}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \mathsf{PI}\left(\right)} + \frac{1}{12} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    5. distribute-rgt-out N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\left(\mathsf{neg}\left(\log \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)\right) - \log f}{\mathsf{PI}\left(\right)}, \left(f \cdot f\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\left(\frac{-1}{1152} \cdot {f}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{12}\right)\right)}\right)\right) \]

    6. lower-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\left(\mathsf{neg}\left(\log \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)\right) - \log f}{\mathsf{PI}\left(\right)}, \left(f \cdot f\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\left(\frac{-1}{1152} \cdot {f}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{12}\right)\right)}\right)\right) \]

    7. lower-PI.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\left(\mathsf{neg}\left(\log \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)\right) - \log f}{\mathsf{PI}\left(\right)}, \left(f \cdot f\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\left(\frac{-1}{1152} \cdot {f}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{12}\right)\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\left(\mathsf{neg}\left(\log \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)\right) - \log f}{\mathsf{PI}\left(\right)}, \left(f \cdot f\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{1152} \cdot {f}^{2}\right)} + \frac{1}{12}\right)\right)\right)\right) \]

    9. lower-fma.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\left(\mathsf{neg}\left(\log \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)\right) - \log f}{\mathsf{PI}\left(\right)}, \left(f \cdot f\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{fma}\left({\mathsf{PI}\left(\right)}^{2}, \frac{-1}{1152} \cdot {f}^{2}, \frac{1}{12}\right)}\right)\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\left(\mathsf{neg}\left(\log \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)\right) - \log f}{\mathsf{PI}\left(\right)}, \left(f \cdot f\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{-1}{1152} \cdot {f}^{2}, \frac{1}{12}\right)\right)\right)\right) \]

    11. lower-*.f64 N/A

        \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\left(\mathsf{neg}\left(\log \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)\right) - \log f}{\mathsf{PI}\left(\right)}, \left(f \cdot f\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{-1}{1152} \cdot {f}^{2}, \frac{1}{12}\right)\right)\right)\right) \]

    12. lower-PI.f64 N/A

        \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\left(\mathsf{neg}\left(\log \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)\right) - \log f}{\mathsf{PI}\left(\right)}, \left(f \cdot f\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), \frac{-1}{1152} \cdot {f}^{2}, \frac{1}{12}\right)\right)\right)\right) \]

    13. lower-PI.f64 N/A

        \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\left(\mathsf{neg}\left(\log \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)\right) - \log f}{\mathsf{PI}\left(\right)}, \left(f \cdot f\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, \frac{-1}{1152} \cdot {f}^{2}, \frac{1}{12}\right)\right)\right)\right) \]

    14. *-commutative N/A

        \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\left(\mathsf{neg}\left(\log \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)\right) - \log f}{\mathsf{PI}\left(\right)}, \left(f \cdot f\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \color{blue}{{f}^{2} \cdot \frac{-1}{1152}}, \frac{1}{12}\right)\right)\right)\right) \]

    15. lower-*.f64 N/A

        \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\left(\mathsf{neg}\left(\log \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)\right) - \log f}{\mathsf{PI}\left(\right)}, \left(f \cdot f\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \color{blue}{{f}^{2} \cdot \frac{-1}{1152}}, \frac{1}{12}\right)\right)\right)\right) \]

    16. unpow2 N/A

        \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(4, \frac{\left(\mathsf{neg}\left(\log \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)\right) - \log f}{\mathsf{PI}\left(\right)}, \left(f \cdot f\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \color{blue}{\left(f \cdot f\right)} \cdot \frac{-1}{1152}, \frac{1}{12}\right)\right)\right)\right) \]

    17. lower-*.f64 97.1

        \[\leadsto -\mathsf{fma}\left(4, \frac{\left(-\log \left(\pi \cdot 0.25\right)\right) - \log f}{\pi}, \left(f \cdot f\right) \cdot \left(\pi \cdot \mathsf{fma}\left(\pi \cdot \pi, \color{blue}{\left(f \cdot f\right)} \cdot -0.0008680555555555555, 0.08333333333333333\right)\right)\right) \]

14. Simplified 97.1%

    \[\leadsto -\mathsf{fma}\left(4, \frac{\left(-\log \left(\pi \cdot 0.25\right)\right) - \log f}{\pi}, \left(f \cdot f\right) \cdot \color{blue}{\left(\pi \cdot \mathsf{fma}\left(\pi \cdot \pi, \left(f \cdot f\right) \cdot -0.0008680555555555555, 0.08333333333333333\right)\right)}\right) \]

15. Final simplification 97.1%

    \[\leadsto -\mathsf{fma}\left(4, \frac{\left(-\log f\right) - \log \left(0.25 \cdot \pi\right)}{\pi}, \left(f \cdot f\right) \cdot \left(\pi \cdot \mathsf{fma}\left(\pi \cdot \pi, \left(f \cdot f\right) \cdot -0.0008680555555555555, 0.08333333333333333\right)\right)\right) \]
16. Add Preprocessing

Alternative 3: 96.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ -\mathsf{fma}\left(\log \left(0.25 \cdot \pi\right) + \log f, \frac{-4}{\pi}, f \cdot \left(f \cdot \left(\pi \cdot 0.08333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (fma
   (+ (log (* 0.25 PI)) (log f))
   (/ -4.0 PI)
   (* f (* f (* PI 0.08333333333333333))))))

C

double code(double f) {
	return -fma((log((0.25 * ((double) M_PI))) + log(f)), (-4.0 / ((double) M_PI)), (f * (f * (((double) M_PI) * 0.08333333333333333))));
}

Julia

function code(f)
	return Float64(-fma(Float64(log(Float64(0.25 * pi)) + log(f)), Float64(-4.0 / pi), Float64(f * Float64(f * Float64(pi * 0.08333333333333333)))))
end

Wolfram

code[f_] := (-N[(N[(N[Log[N[(0.25 * Pi), $MachinePrecision]], $MachinePrecision] + N[Log[f], $MachinePrecision]), $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision] + N[(f * N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in f around 0

   \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{f \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + f \cdot \left(\frac{1}{16} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}}{{\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)\right) + 2 \cdot \frac{1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{f}\right)}\right) \]

5. Simplified 97.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(f, f \cdot \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \pi \cdot 0.125\right), \frac{4}{\pi}\right)}{f}\right)} \]

6. Taylor expanded in f around 0

   \[\leadsto \mathsf{neg}\left(\color{blue}{\left(4 \cdot \frac{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f}{\mathsf{PI}\left(\right)} + {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\frac{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f}{\mathsf{PI}\left(\right)} \cdot 4} + {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\frac{\left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f\right) \cdot 4}{\mathsf{PI}\left(\right)}} + {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f\right) \cdot \frac{4}{\mathsf{PI}\left(\right)}} + {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f, \frac{4}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + \color{blue}{\left(\mathsf{neg}\left(\log f\right)\right)}, \frac{4}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\color{blue}{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \log f}, \frac{4}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. lower--.f64 N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\color{blue}{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \log f}, \frac{4}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. lower-log.f64 N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\color{blue}{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right)} - \log f, \frac{4}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\log \color{blue}{\left(\frac{4}{\mathsf{PI}\left(\right)}\right)} - \log f, \frac{4}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   10. lower-PI.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\log \left(\frac{4}{\color{blue}{\mathsf{PI}\left(\right)}}\right) - \log f, \frac{4}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   11. lower-log.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \color{blue}{\log f}, \frac{4}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   12. lower-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \log f, \color{blue}{\frac{4}{\mathsf{PI}\left(\right)}}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   13. lower-PI.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \log f, \frac{4}{\color{blue}{\mathsf{PI}\left(\right)}}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \log f, \frac{4}{\mathsf{PI}\left(\right)}, \color{blue}{\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right) \cdot {f}^{2}}\right)\right) \]

8. Simplified 97.0%

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

9. Taylor expanded in f around inf

   \[\leadsto \mathsf{neg}\left(\color{blue}{{f}^{2} \cdot \left(\frac{1}{12} \cdot \mathsf{PI}\left(\right) + 4 \cdot \frac{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - -1 \cdot \log \left(\frac{1}{f}\right)}{{f}^{2} \cdot \mathsf{PI}\left(\right)}\right)}\right) \]

11. Simplified 47.6%

    \[\leadsto -\color{blue}{\left(f \cdot f\right) \cdot \mathsf{fma}\left(\left(-\log \left(\pi \cdot 0.25\right)\right) - \log f, \frac{4}{\pi \cdot \left(f \cdot f\right)}, 0.08333333333333333 \cdot \pi\right)} \]

12. Taylor expanded in f around 0

    \[\leadsto \mathsf{neg}\left(\color{blue}{\left(-4 \cdot \frac{\log f + \log \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right)} + \frac{1}{12} \cdot \left({f}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
13. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\frac{-4 \cdot \left(\log f + \log \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}{\mathsf{PI}\left(\right)}} + \frac{1}{12} \cdot \left({f}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\left(\frac{\color{blue}{\left(\log f + \log \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot -4}}{\mathsf{PI}\left(\right)} + \frac{1}{12} \cdot \left({f}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    3. remove-double-neg N/A

       \[\leadsto \mathsf{neg}\left(\left(\frac{\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log f\right)\right)\right)\right)} + \log \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot -4}{\mathsf{PI}\left(\right)} + \frac{1}{12} \cdot \left({f}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    4. mul-1-neg N/A

       \[\leadsto \mathsf{neg}\left(\left(\frac{\left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \log f}\right)\right) + \log \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot -4}{\mathsf{PI}\left(\right)} + \frac{1}{12} \cdot \left({f}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    5. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{neg}\left(\left(\frac{\left(\color{blue}{-1 \cdot \left(\mathsf{neg}\left(\log f\right)\right)} + \log \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot -4}{\mathsf{PI}\left(\right)} + \frac{1}{12} \cdot \left({f}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    6. log-rec N/A

       \[\leadsto \mathsf{neg}\left(\left(\frac{\left(-1 \cdot \color{blue}{\log \left(\frac{1}{f}\right)} + \log \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot -4}{\mathsf{PI}\left(\right)} + \frac{1}{12} \cdot \left({f}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    7. +-commutative N/A

       \[\leadsto \mathsf{neg}\left(\left(\frac{\color{blue}{\left(\log \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) + -1 \cdot \log \left(\frac{1}{f}\right)\right)} \cdot -4}{\mathsf{PI}\left(\right)} + \frac{1}{12} \cdot \left({f}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    8. associate-/l* N/A

       \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\log \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) + -1 \cdot \log \left(\frac{1}{f}\right)\right) \cdot \frac{-4}{\mathsf{PI}\left(\right)}} + \frac{1}{12} \cdot \left({f}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\left(\left(\log \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) + -1 \cdot \log \left(\frac{1}{f}\right)\right) \cdot \frac{-4}{\mathsf{PI}\left(\right)} + \frac{1}{12} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {f}^{2}\right)}\right)\right) \]

    10. associate-*r* N/A

        \[\leadsto \mathsf{neg}\left(\left(\left(\log \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) + -1 \cdot \log \left(\frac{1}{f}\right)\right) \cdot \frac{-4}{\mathsf{PI}\left(\right)} + \color{blue}{\left(\frac{1}{12} \cdot \mathsf{PI}\left(\right)\right) \cdot {f}^{2}}\right)\right) \]

    11. lower-fma.f64 N/A

        \[\leadsto \mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(\log \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) + -1 \cdot \log \left(\frac{1}{f}\right), \frac{-4}{\mathsf{PI}\left(\right)}, \left(\frac{1}{12} \cdot \mathsf{PI}\left(\right)\right) \cdot {f}^{2}\right)}\right) \]

14. Simplified 97.0%

    \[\leadsto -\color{blue}{\mathsf{fma}\left(\log f + \log \left(\pi \cdot 0.25\right), \frac{-4}{\pi}, f \cdot \left(f \cdot \left(\pi \cdot 0.08333333333333333\right)\right)\right)} \]

15. Final simplification 97.0%

    \[\leadsto -\mathsf{fma}\left(\log \left(0.25 \cdot \pi\right) + \log f, \frac{-4}{\pi}, f \cdot \left(f \cdot \left(\pi \cdot 0.08333333333333333\right)\right)\right) \]
16. Add Preprocessing

Alternative 4: 96.2% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in f around 0

   \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{f \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + f \cdot \left(\frac{1}{16} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}}{{\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)\right) + 2 \cdot \frac{1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{f}\right)}\right) \]

5. Simplified 97.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(f, f \cdot \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \pi \cdot 0.125\right), \frac{4}{\pi}\right)}{f}\right)} \]

6. Taylor expanded in f around 0

   \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f, \color{blue}{f \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)}, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right)\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f, \color{blue}{\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right) \cdot f}, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right)\right) \]

   2. distribute-rgt-out N/A

      \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{24} + \frac{1}{8}\right)\right)} \cdot f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f, \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\left(\frac{-1}{24} + \frac{1}{8}\right) \cdot f\right)}, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f, \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\left(\frac{-1}{24} + \frac{1}{8}\right) \cdot f\right)}, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right)\right) \]

   5. lower-PI.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f, \color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\left(\frac{-1}{24} + \frac{1}{8}\right) \cdot f\right), \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(f, \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\left(\frac{-1}{24} + \frac{1}{8}\right) \cdot f\right)}, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right)\right) \]

   7. metadata-eval 97.0

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

8. Simplified 97.0%

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

9. Final simplification 97.0%

   \[\leadsto \log \left(\frac{\mathsf{fma}\left(f, \pi \cdot \left(f \cdot 0.08333333333333333\right), \frac{4}{\pi}\right)}{f}\right) \cdot \frac{-1}{\frac{\pi}{4}} \]
10. Add Preprocessing

Alternative 5: 95.7% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in f around inf

   \[\leadsto \mathsf{neg}\left(\color{blue}{4 \cdot \frac{\log \left(\frac{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)}}\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \frac{\color{blue}{\frac{2}{\frac{1}{2}}}}{\mathsf{PI}\left(\right)}\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \frac{\frac{2}{\color{blue}{\frac{1}{4} - \frac{-1}{4}}}}{\mathsf{PI}\left(\right)}\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \color{blue}{\frac{2}{\left(\frac{1}{4} - \frac{-1}{4}\right) \cdot \mathsf{PI}\left(\right)}}\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \frac{2}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)}}\right) \]

   8. distribute-rgt-out-- N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \frac{2}{\color{blue}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \frac{\color{blue}{2 \cdot 1}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \color{blue}{\left(2 \cdot \frac{1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}\right)}\right) \]

5. Simplified 7.5%

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

6. Taylor expanded in f around 0

   \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(\frac{2}{f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}\right)} \cdot \frac{4}{\mathsf{PI}\left(\right)}\right) \]
7. Step-by-step derivation

   1. associate-/l/ N/A

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

   2. distribute-rgt-out-- N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{\frac{2}{\color{blue}{\frac{1}{2}}}}{\mathsf{PI}\left(\right)}}{f}\right) \cdot \frac{4}{\mathsf{PI}\left(\right)}\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{\color{blue}{4}}{\mathsf{PI}\left(\right)}}{f}\right) \cdot \frac{4}{\mathsf{PI}\left(\right)}\right) \]

   7. associate-/l/ N/A

      \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(\frac{4}{f \cdot \mathsf{PI}\left(\right)}\right)} \cdot \frac{4}{\mathsf{PI}\left(\right)}\right) \]

   8. lower-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(\frac{4}{f \cdot \mathsf{PI}\left(\right)}\right)} \cdot \frac{4}{\mathsf{PI}\left(\right)}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{4}{\color{blue}{\mathsf{PI}\left(\right) \cdot f}}\right) \cdot \frac{4}{\mathsf{PI}\left(\right)}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\log \left(\frac{4}{\color{blue}{\mathsf{PI}\left(\right) \cdot f}}\right) \cdot \frac{4}{\mathsf{PI}\left(\right)}\right) \]

   11. lower-PI.f64 96.7

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

8. Simplified 96.7%

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

9. Final simplification 96.7%

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

Alternative 6: 4.2% accurate, 38.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in f around 0

   \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{f \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + f \cdot \left(\frac{1}{16} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}}{{\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)\right) + 2 \cdot \frac{1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{f}\right)}\right) \]

5. Simplified 97.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(f, f \cdot \mathsf{fma}\left(\pi \cdot 0.020833333333333332, -2, \pi \cdot 0.125\right), \frac{4}{\pi}\right)}{f}\right)} \]

6. Taylor expanded in f around 0

   \[\leadsto \mathsf{neg}\left(\color{blue}{\left(4 \cdot \frac{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f}{\mathsf{PI}\left(\right)} + {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\frac{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f}{\mathsf{PI}\left(\right)} \cdot 4} + {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\frac{\left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f\right) \cdot 4}{\mathsf{PI}\left(\right)}} + {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f\right) \cdot \frac{4}{\mathsf{PI}\left(\right)}} + {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f, \frac{4}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + \color{blue}{\left(\mathsf{neg}\left(\log f\right)\right)}, \frac{4}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\color{blue}{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \log f}, \frac{4}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. lower--.f64 N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\color{blue}{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \log f}, \frac{4}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. lower-log.f64 N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\color{blue}{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right)} - \log f, \frac{4}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\log \color{blue}{\left(\frac{4}{\mathsf{PI}\left(\right)}\right)} - \log f, \frac{4}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   10. lower-PI.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\log \left(\frac{4}{\color{blue}{\mathsf{PI}\left(\right)}}\right) - \log f, \frac{4}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   11. lower-log.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \color{blue}{\log f}, \frac{4}{\mathsf{PI}\left(\right)}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   12. lower-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \log f, \color{blue}{\frac{4}{\mathsf{PI}\left(\right)}}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   13. lower-PI.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \log f, \frac{4}{\color{blue}{\mathsf{PI}\left(\right)}}, {f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \log f, \frac{4}{\mathsf{PI}\left(\right)}, \color{blue}{\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right) \cdot {f}^{2}}\right)\right) \]

8. Simplified 97.0%

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

9. Taylor expanded in f around inf

   \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{1}{12} \cdot \left({f}^{2} \cdot \mathsf{PI}\left(\right)\right)}\right) \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{1}{12} \cdot {f}^{2}\right) \cdot \mathsf{PI}\left(\right)}\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{12} \cdot {f}^{2}\right)}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{12} \cdot {f}^{2}\right)}\right) \]

    4. lower-PI.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{12} \cdot {f}^{2}\right)\right) \]

    5. lower-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{12} \cdot {f}^{2}\right)}\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{12} \cdot \color{blue}{\left(f \cdot f\right)}\right)\right) \]

    7. lower-*.f64 4.3

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

11. Simplified 4.3%

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

12. Taylor expanded in f around 0

    \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({f}^{2} \cdot \mathsf{PI}\left(\right)\right)} \]
13. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\frac{-1}{12} \cdot {f}^{2}\right) \cdot \mathsf{PI}\left(\right)} \]

    2. *-commutative N/A

       \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{12} \cdot {f}^{2}\right)} \]

    3. lower-*.f64 N/A

       \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{12} \cdot {f}^{2}\right)} \]

    4. lower-PI.f64 N/A

       \[\leadsto \color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{-1}{12} \cdot {f}^{2}\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{\left(f \cdot f\right)}\right) \]

    6. associate-*r* N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\left(\frac{-1}{12} \cdot f\right) \cdot f\right)} \]

    7. *-commutative N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\left(f \cdot \left(\frac{-1}{12} \cdot f\right)\right)} \]

    8. lower-*.f64 N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\left(f \cdot \left(\frac{-1}{12} \cdot f\right)\right)} \]

    9. *-commutative N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \left(f \cdot \color{blue}{\left(f \cdot \frac{-1}{12}\right)}\right) \]

    10. lower-*.f64 4.3

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

14. Simplified 4.3%

    \[\leadsto \color{blue}{\pi \cdot \left(f \cdot \left(f \cdot -0.08333333333333333\right)\right)} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(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)))))))))