VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 97.2%

Time: 32.2s

Alternatives: 11

Speedup: N/A×

• could not determine a ground truth

  1. f = 1560559565.1241376

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: 97.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 \cdot \log 2\\ t_1 := f \cdot \frac{\pi}{4}\\ \frac{\left(t\_0 - \log \cosh t\_1\right) - \left(t\_0 - \log \sinh t\_1\right)}{\frac{\pi}{4}} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* -1.0 (log 2.0))) (t_1 (* f (/ PI 4.0))))
   (/ (- (- t_0 (log (cosh t_1))) (- t_0 (log (sinh t_1)))) (/ PI 4.0))))

C

double code(double f) {
	double t_0 = -1.0 * log(2.0);
	double t_1 = f * (((double) M_PI) / 4.0);
	return ((t_0 - log(cosh(t_1))) - (t_0 - log(sinh(t_1)))) / (((double) M_PI) / 4.0);
}

Java

public static double code(double f) {
	double t_0 = -1.0 * Math.log(2.0);
	double t_1 = f * (Math.PI / 4.0);
	return ((t_0 - Math.log(Math.cosh(t_1))) - (t_0 - Math.log(Math.sinh(t_1)))) / (Math.PI / 4.0);
}

Python

def code(f):
	t_0 = -1.0 * math.log(2.0)
	t_1 = f * (math.pi / 4.0)
	return ((t_0 - math.log(math.cosh(t_1))) - (t_0 - math.log(math.sinh(t_1)))) / (math.pi / 4.0)

Julia

function code(f)
	t_0 = Float64(-1.0 * log(2.0))
	t_1 = Float64(f * Float64(pi / 4.0))
	return Float64(Float64(Float64(t_0 - log(cosh(t_1))) - Float64(t_0 - log(sinh(t_1)))) / Float64(pi / 4.0))
end

MATLAB

function tmp = code(f)
	t_0 = -1.0 * log(2.0);
	t_1 = f * (pi / 4.0);
	tmp = ((t_0 - log(cosh(t_1))) - (t_0 - log(sinh(t_1)))) / (pi / 4.0);
end

Wolfram

code[f_] := Block[{t$95$0 = N[(-1.0 * N[Log[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(f * N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(t$95$0 - N[Log[N[Cosh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 - N[Log[N[Sinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 8.1%

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

4. Applied rewrites 97.5%

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

5. Final simplification 97.5%

   \[\leadsto \frac{\left(-1 \cdot \log 2 - \log \cosh \left(f \cdot \frac{\pi}{4}\right)\right) - \left(-1 \cdot \log 2 - \log \sinh \left(f \cdot \frac{\pi}{4}\right)\right)}{\frac{\pi}{4}} \]
6. Add Preprocessing

Alternative 2: 97.1% accurate, N/A× speedup

Math

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

FPCore

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

C

double code(double f) {
	return (((-1.0 * log(2.0)) - log(cosh((f * (((double) M_PI) / 4.0))))) + log((sinh(((((double) M_PI) / 4.0) * f)) * 2.0))) / (((double) M_PI) / 4.0);
}

Java

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

Python

def code(f):
	return (((-1.0 * math.log(2.0)) - math.log(math.cosh((f * (math.pi / 4.0))))) + math.log((math.sinh(((math.pi / 4.0) * f)) * 2.0))) / (math.pi / 4.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.1%

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

4. Applied rewrites 97.5%

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

   1. lift-+.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. lift-sinh.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. +-commutative N/A

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

   9. sum-log N/A

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

   10. lower-log.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-sinh.f64 97.4

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 97.4

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

6. Applied rewrites 97.4%

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

7. Final simplification 97.4%

   \[\leadsto \frac{\left(-1 \cdot \log 2 - \log \cosh \left(f \cdot \frac{\pi}{4}\right)\right) + \log \left(\sinh \left(\frac{\pi}{4} \cdot f\right) \cdot 2\right)}{\frac{\pi}{4}} \]
8. Add Preprocessing

Alternative 3: 97.0% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.1%

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

   1. lift--.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. sinh-undef N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sinh.f64 97.0

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 97.0

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

4. Applied rewrites 97.0%

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

5. Final simplification 97.0%

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

Alternative 4: 96.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot \pi\right) \cdot \pi\\ t_1 := t\_0 \cdot 0.005208333333333333\\ t_2 := \frac{\pi}{\pi \cdot 0.5} \cdot 0\\ t_3 := {\left(\pi \cdot 0.5\right)}^{2}\\ t_4 := \frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(t\_0, 0.005208333333333333, t\_1\right)}{t\_3}\\ t_5 := {t\_2}^{2}\\ \left(-1 \cdot \frac{1}{\frac{\pi}{4}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot f, \mathsf{fma}\left(-1.5 \cdot t\_2, t\_4 \cdot t\_3, \mathsf{fma}\left(3 \cdot \left(\pi \cdot 0.5\right), \frac{t\_0 \cdot 0}{\pi \cdot 0.5} - \left(-1 \cdot \frac{t\_1}{\pi}\right) \cdot \frac{t\_2}{0.5}, \left(0.25 \cdot {t\_5}^{1.5}\right) \cdot \left(t\_0 \cdot 0.125\right)\right)\right), 0.5 \cdot \mathsf{fma}\left(t\_4, \pi \cdot 0.5, \left(-0.25 \cdot t\_5\right) \cdot t\_3\right)\right), f, \left(0.5 \cdot t\_2\right) \cdot \left(\pi \cdot 0.5\right)\right), f, \log f \cdot -1\right) + \log \left({\left(\pi \cdot 0.5\right)}^{-1} \cdot 2\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (* PI PI) PI))
        (t_1 (* t_0 0.005208333333333333))
        (t_2 (* (/ PI (* PI 0.5)) 0.0))
        (t_3 (pow (* PI 0.5) 2.0))
        (t_4
         (-
          (* (/ (* PI PI) PI) 0.125)
          (/ (fma t_0 0.005208333333333333 t_1) t_3)))
        (t_5 (pow t_2 2.0)))
   (*
    (* -1.0 (/ 1.0 (/ PI 4.0)))
    (+
     (fma
      (fma
       (fma
        (* 0.16666666666666666 f)
        (fma
         (* -1.5 t_2)
         (* t_4 t_3)
         (fma
          (* 3.0 (* PI 0.5))
          (- (/ (* t_0 0.0) (* PI 0.5)) (* (* -1.0 (/ t_1 PI)) (/ t_2 0.5)))
          (* (* 0.25 (pow t_5 1.5)) (* t_0 0.125))))
        (* 0.5 (fma t_4 (* PI 0.5) (* (* -0.25 t_5) t_3))))
       f
       (* (* 0.5 t_2) (* PI 0.5)))
      f
      (* (log f) -1.0))
     (log (* (pow (* PI 0.5) -1.0) 2.0))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) * ((double) M_PI)) * ((double) M_PI);
	double t_1 = t_0 * 0.005208333333333333;
	double t_2 = (((double) M_PI) / (((double) M_PI) * 0.5)) * 0.0;
	double t_3 = pow((((double) M_PI) * 0.5), 2.0);
	double t_4 = (((((double) M_PI) * ((double) M_PI)) / ((double) M_PI)) * 0.125) - (fma(t_0, 0.005208333333333333, t_1) / t_3);
	double t_5 = pow(t_2, 2.0);
	return (-1.0 * (1.0 / (((double) M_PI) / 4.0))) * (fma(fma(fma((0.16666666666666666 * f), fma((-1.5 * t_2), (t_4 * t_3), fma((3.0 * (((double) M_PI) * 0.5)), (((t_0 * 0.0) / (((double) M_PI) * 0.5)) - ((-1.0 * (t_1 / ((double) M_PI))) * (t_2 / 0.5))), ((0.25 * pow(t_5, 1.5)) * (t_0 * 0.125)))), (0.5 * fma(t_4, (((double) M_PI) * 0.5), ((-0.25 * t_5) * t_3)))), f, ((0.5 * t_2) * (((double) M_PI) * 0.5))), f, (log(f) * -1.0)) + log((pow((((double) M_PI) * 0.5), -1.0) * 2.0)));
}

Julia

function code(f)
	t_0 = Float64(Float64(pi * pi) * pi)
	t_1 = Float64(t_0 * 0.005208333333333333)
	t_2 = Float64(Float64(pi / Float64(pi * 0.5)) * 0.0)
	t_3 = Float64(pi * 0.5) ^ 2.0
	t_4 = Float64(Float64(Float64(Float64(pi * pi) / pi) * 0.125) - Float64(fma(t_0, 0.005208333333333333, t_1) / t_3))
	t_5 = t_2 ^ 2.0
	return Float64(Float64(-1.0 * Float64(1.0 / Float64(pi / 4.0))) * Float64(fma(fma(fma(Float64(0.16666666666666666 * f), fma(Float64(-1.5 * t_2), Float64(t_4 * t_3), fma(Float64(3.0 * Float64(pi * 0.5)), Float64(Float64(Float64(t_0 * 0.0) / Float64(pi * 0.5)) - Float64(Float64(-1.0 * Float64(t_1 / pi)) * Float64(t_2 / 0.5))), Float64(Float64(0.25 * (t_5 ^ 1.5)) * Float64(t_0 * 0.125)))), Float64(0.5 * fma(t_4, Float64(pi * 0.5), Float64(Float64(-0.25 * t_5) * t_3)))), f, Float64(Float64(0.5 * t_2) * Float64(pi * 0.5))), f, Float64(log(f) * -1.0)) + log(Float64((Float64(pi * 0.5) ^ -1.0) * 2.0))))
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 0.005208333333333333), $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(Pi * 0.5), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(Pi * Pi), $MachinePrecision] / Pi), $MachinePrecision] * 0.125), $MachinePrecision] - N[(N[(t$95$0 * 0.005208333333333333 + t$95$1), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[t$95$2, 2.0], $MachinePrecision]}, N[(N[(-1.0 * N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(0.16666666666666666 * f), $MachinePrecision] * N[(N[(-1.5 * t$95$2), $MachinePrecision] * N[(t$95$4 * t$95$3), $MachinePrecision] + N[(N[(3.0 * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 * 0.0), $MachinePrecision] / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 * N[(t$95$1 / Pi), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.25 * N[Power[t$95$5, 1.5], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$4 * N[(Pi * 0.5), $MachinePrecision] + N[(N[(-0.25 * t$95$5), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * f + N[(N[(0.5 * t$95$2), $MachinePrecision] * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * f + N[(N[Log[f], $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] + N[Log[N[(N[Power[N[(Pi * 0.5), $MachinePrecision], -1.0], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 8.1%

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

3. Taylor expanded in f around 0

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\frac{2}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}\right) + \left(-1 \cdot \log f + f \cdot \left(\frac{1}{2} \cdot \left(\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)} + \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)}\right) \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) + f \cdot \left(\frac{1}{6} \cdot \left(f \cdot \left(\frac{-3}{2} \cdot \left(\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)} + \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)}\right) \cdot \left(\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) \cdot {\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right) + \left(\frac{1}{4} \cdot \left({\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)} + \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)}\right)}^{3} \cdot {\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{3}\right) + 3 \cdot \left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\frac{-1}{384} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + \frac{1}{384} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}\right) - \frac{\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)} + \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)}\right) \cdot \left(\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\frac{-1}{4} \cdot \left({\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)} + \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)}\right)}^{2} \cdot {\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \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) \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)} \]

5. Applied rewrites 96.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot f, \mathsf{fma}\left(-1.5 \cdot \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot 0.5\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot 0.5} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot 0.5} \cdot 0}{0.5}, \left(0.25 \cdot {\left({\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right)}^{1.5}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.125\right)\right)\right), 0.5 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \pi \cdot 0.5, \left(-0.25 \cdot {\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}\right)\right), f, \left(0.5 \cdot \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)\right) \cdot \left(\pi \cdot 0.5\right)\right), f, \log f \cdot -1\right) + \log \left({\left(\pi \cdot 0.5\right)}^{-1} \cdot 2\right)\right)} \]

6. Final simplification 96.6%

   \[\leadsto \left(-1 \cdot \frac{1}{\frac{\pi}{4}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot f, \mathsf{fma}\left(-1.5 \cdot \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot 0.5\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot 0.5} - \left(-1 \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333}{\pi}\right) \cdot \frac{\frac{\pi}{\pi \cdot 0.5} \cdot 0}{0.5}, \left(0.25 \cdot {\left({\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right)}^{1.5}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.125\right)\right)\right), 0.5 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \pi \cdot 0.5, \left(-0.25 \cdot {\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}\right)\right), f, \left(0.5 \cdot \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)\right) \cdot \left(\pi \cdot 0.5\right)\right), f, \log f \cdot -1\right) + \log \left({\left(\pi \cdot 0.5\right)}^{-1} \cdot 2\right)\right) \]
7. Add Preprocessing

Alternative 5: 96.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi \cdot 0.5}{\pi}\\ t_1 := \left(\pi \cdot \pi\right) \cdot \pi\\ t_2 := \frac{\pi}{\pi \cdot 0.5} \cdot 0\\ t_3 := t\_1 \cdot 0.005208333333333333\\ t_4 := {\left(\pi \cdot 0.5\right)}^{2}\\ t_5 := \frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(t\_1, 0.005208333333333333, t\_3\right)}{t\_4}\\ t_6 := {t\_2}^{2}\\ -1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, \mathsf{fma}\left(-1.5 \cdot t\_2, t\_5 \cdot t\_4, \mathsf{fma}\left(3 \cdot \left(\pi \cdot 0.5\right), \frac{t\_1 \cdot 0}{\pi \cdot 0.5} - \left(-1 \cdot \frac{t\_3}{\pi}\right) \cdot \frac{t\_2}{0.5}, \left(0.25 \cdot {t\_6}^{1.5}\right) \cdot \left(t\_1 \cdot 0.125\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(t\_5, \pi \cdot 0.5, \left(-0.25 \cdot t\_6\right) \cdot t\_4\right)\right)}{\pi}, f, \mathsf{fma}\left(t\_2, t\_0, t\_2 \cdot t\_0\right)\right), f, \frac{\frac{4}{{\pi}^{0.5}} \cdot \mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ (* PI 0.5) PI))
        (t_1 (* (* PI PI) PI))
        (t_2 (* (/ PI (* PI 0.5)) 0.0))
        (t_3 (* t_1 0.005208333333333333))
        (t_4 (pow (* PI 0.5) 2.0))
        (t_5
         (-
          (* (/ (* PI PI) PI) 0.125)
          (/ (fma t_1 0.005208333333333333 t_3) t_4)))
        (t_6 (pow t_2 2.0)))
   (*
    -1.0
    (fma
     (fma
      (/
       (fma
        0.6666666666666666
        (*
         (fma
          (* -1.5 t_2)
          (* t_5 t_4)
          (fma
           (* 3.0 (* PI 0.5))
           (- (/ (* t_1 0.0) (* PI 0.5)) (* (* -1.0 (/ t_3 PI)) (/ t_2 0.5)))
           (* (* 0.25 (pow t_6 1.5)) (* t_1 0.125))))
         f)
        (* 2.0 (fma t_5 (* PI 0.5) (* (* -0.25 t_6) t_4))))
       PI)
      f
      (fma t_2 t_0 (* t_2 t_0)))
     f
     (/
      (* (/ 4.0 (pow PI 0.5)) (fma (log f) -1.0 (log (/ 2.0 (* 0.5 PI)))))
      (pow PI 0.5))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) * 0.5) / ((double) M_PI);
	double t_1 = (((double) M_PI) * ((double) M_PI)) * ((double) M_PI);
	double t_2 = (((double) M_PI) / (((double) M_PI) * 0.5)) * 0.0;
	double t_3 = t_1 * 0.005208333333333333;
	double t_4 = pow((((double) M_PI) * 0.5), 2.0);
	double t_5 = (((((double) M_PI) * ((double) M_PI)) / ((double) M_PI)) * 0.125) - (fma(t_1, 0.005208333333333333, t_3) / t_4);
	double t_6 = pow(t_2, 2.0);
	return -1.0 * fma(fma((fma(0.6666666666666666, (fma((-1.5 * t_2), (t_5 * t_4), fma((3.0 * (((double) M_PI) * 0.5)), (((t_1 * 0.0) / (((double) M_PI) * 0.5)) - ((-1.0 * (t_3 / ((double) M_PI))) * (t_2 / 0.5))), ((0.25 * pow(t_6, 1.5)) * (t_1 * 0.125)))) * f), (2.0 * fma(t_5, (((double) M_PI) * 0.5), ((-0.25 * t_6) * t_4)))) / ((double) M_PI)), f, fma(t_2, t_0, (t_2 * t_0))), f, (((4.0 / pow(((double) M_PI), 0.5)) * fma(log(f), -1.0, log((2.0 / (0.5 * ((double) M_PI)))))) / pow(((double) M_PI), 0.5)));
}

Julia

function code(f)
	t_0 = Float64(Float64(pi * 0.5) / pi)
	t_1 = Float64(Float64(pi * pi) * pi)
	t_2 = Float64(Float64(pi / Float64(pi * 0.5)) * 0.0)
	t_3 = Float64(t_1 * 0.005208333333333333)
	t_4 = Float64(pi * 0.5) ^ 2.0
	t_5 = Float64(Float64(Float64(Float64(pi * pi) / pi) * 0.125) - Float64(fma(t_1, 0.005208333333333333, t_3) / t_4))
	t_6 = t_2 ^ 2.0
	return Float64(-1.0 * fma(fma(Float64(fma(0.6666666666666666, Float64(fma(Float64(-1.5 * t_2), Float64(t_5 * t_4), fma(Float64(3.0 * Float64(pi * 0.5)), Float64(Float64(Float64(t_1 * 0.0) / Float64(pi * 0.5)) - Float64(Float64(-1.0 * Float64(t_3 / pi)) * Float64(t_2 / 0.5))), Float64(Float64(0.25 * (t_6 ^ 1.5)) * Float64(t_1 * 0.125)))) * f), Float64(2.0 * fma(t_5, Float64(pi * 0.5), Float64(Float64(-0.25 * t_6) * t_4)))) / pi), f, fma(t_2, t_0, Float64(t_2 * t_0))), f, Float64(Float64(Float64(4.0 / (pi ^ 0.5)) * fma(log(f), -1.0, log(Float64(2.0 / Float64(0.5 * pi))))) / (pi ^ 0.5))))
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi * 0.5), $MachinePrecision] / Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * 0.005208333333333333), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(Pi * 0.5), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(Pi * Pi), $MachinePrecision] / Pi), $MachinePrecision] * 0.125), $MachinePrecision] - N[(N[(t$95$1 * 0.005208333333333333 + t$95$3), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[t$95$2, 2.0], $MachinePrecision]}, N[(-1.0 * N[(N[(N[(N[(0.6666666666666666 * N[(N[(N[(-1.5 * t$95$2), $MachinePrecision] * N[(t$95$5 * t$95$4), $MachinePrecision] + N[(N[(3.0 * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$1 * 0.0), $MachinePrecision] / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 * N[(t$95$3 / Pi), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.25 * N[Power[t$95$6, 1.5], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * f), $MachinePrecision] + N[(2.0 * N[(t$95$5 * N[(Pi * 0.5), $MachinePrecision] + N[(N[(-0.25 * t$95$6), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * f + N[(t$95$2 * t$95$0 + N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * f + N[(N[(N[(4.0 / N[Power[Pi, 0.5], $MachinePrecision]), $MachinePrecision] * N[(N[Log[f], $MachinePrecision] * -1.0 + N[Log[N[(2.0 / N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[Pi, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 8.1%

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

3. Taylor expanded in f around 0

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

5. Applied rewrites 96.5%

   \[\leadsto -\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, \mathsf{fma}\left(-1.5 \cdot \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot 0.5\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot 0.5} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot 0.5} \cdot 0}{0.5}, \left(0.25 \cdot {\left({\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right)}^{1.5}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.125\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \pi \cdot 0.5, \left(-0.25 \cdot {\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0, \frac{\pi \cdot 0.5}{\pi}, \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right) \cdot \frac{\pi \cdot 0.5}{\pi}\right)\right), f, \frac{4}{{\pi}^{0.5}} \cdot \frac{\mathsf{fma}\left(\log f, -1, \log \left({\left(\pi \cdot 0.5\right)}^{-1} \cdot 2\right)\right)}{{\pi}^{0.5}}\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \frac{\mathsf{fma}\left(\log f, -1, \log \left({\left(\pi \cdot \frac{1}{2}\right)}^{-1} \cdot 2\right)\right)}{{\pi}^{\frac{1}{2}}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \frac{\mathsf{fma}\left(\log f, -1, \log \left({\left(\pi \cdot \frac{1}{2}\right)}^{-1} \cdot 2\right)\right)}{{\pi}^{\frac{1}{2}}}\right) \]

   3. lift-log.f64 N/A

      \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \frac{\mathsf{fma}\left(\log f, -1, \log \left({\left(\pi \cdot \frac{1}{2}\right)}^{-1} \cdot 2\right)\right)}{{\pi}^{\frac{1}{2}}}\right) \]

   4. lift-fma.f64 N/A

      \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \frac{\log f \cdot -1 + \log \left({\left(\pi \cdot \frac{1}{2}\right)}^{-1} \cdot 2\right)}{{\pi}^{\frac{1}{2}}}\right) \]

   5. lift-log.f64 N/A

      \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \frac{\log f \cdot -1 + \log \left({\left(\pi \cdot \frac{1}{2}\right)}^{-1} \cdot 2\right)}{{\pi}^{\frac{1}{2}}}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \frac{\log f \cdot -1 + \log \left({\left(\pi \cdot \frac{1}{2}\right)}^{-1} \cdot 2\right)}{{\pi}^{\frac{1}{2}}}\right) \]

   7. lift-pow.f64 N/A

      \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \frac{\log f \cdot -1 + \log \left({\left(\pi \cdot \frac{1}{2}\right)}^{-1} \cdot 2\right)}{{\pi}^{\frac{1}{2}}}\right) \]

   8. lift-PI.f64 N/A

      \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \frac{\log f \cdot -1 + \log \left({\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)}^{-1} \cdot 2\right)}{{\pi}^{\frac{1}{2}}}\right) \]

   9. lift-*.f64 N/A

      \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \frac{\log f \cdot -1 + \log \left({\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)}^{-1} \cdot 2\right)}{{\pi}^{\frac{1}{2}}}\right) \]

   10. associate-*r/ N/A

       \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{\frac{4}{{\pi}^{\frac{1}{2}}} \cdot \left(\log f \cdot -1 + \log \left({\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)}^{-1} \cdot 2\right)\right)}{{\pi}^{\frac{1}{2}}}\right) \]

7. Applied rewrites 96.6%

   \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, \mathsf{fma}\left(-1.5 \cdot \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot 0.5\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot 0.5} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot 0.5} \cdot 0}{0.5}, \left(0.25 \cdot {\left({\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right)}^{1.5}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.125\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \pi \cdot 0.5, \left(-0.25 \cdot {\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0, \frac{\pi \cdot 0.5}{\pi}, \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right) \cdot \frac{\pi \cdot 0.5}{\pi}\right)\right), f, \frac{\frac{4}{{\pi}^{0.5}} \cdot \mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}}\right) \]

8. Final simplification 96.6%

   \[\leadsto -1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, \mathsf{fma}\left(-1.5 \cdot \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot 0.5\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot 0.5} - \left(-1 \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333}{\pi}\right) \cdot \frac{\frac{\pi}{\pi \cdot 0.5} \cdot 0}{0.5}, \left(0.25 \cdot {\left({\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right)}^{1.5}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.125\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \pi \cdot 0.5, \left(-0.25 \cdot {\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0, \frac{\pi \cdot 0.5}{\pi}, \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right) \cdot \frac{\pi \cdot 0.5}{\pi}\right)\right), f, \frac{\frac{4}{{\pi}^{0.5}} \cdot \mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}}\right) \]
9. Add Preprocessing

Alternative 6: 96.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi \cdot 0.5}{\pi}\\ t_1 := \left(\pi \cdot \pi\right) \cdot \pi\\ t_2 := \frac{\pi}{\pi \cdot 0.5} \cdot 0\\ t_3 := t\_1 \cdot 0.005208333333333333\\ t_4 := {\left(\pi \cdot 0.5\right)}^{2}\\ t_5 := \frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(t\_1, 0.005208333333333333, t\_3\right)}{t\_4}\\ t_6 := {t\_2}^{2}\\ -1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, \mathsf{fma}\left(-1.5 \cdot t\_2, t\_5 \cdot t\_4, \mathsf{fma}\left(3 \cdot \left(\pi \cdot 0.5\right), \frac{t\_1 \cdot 0}{\pi \cdot 0.5} - \left(-1 \cdot \frac{t\_3}{\pi}\right) \cdot \frac{t\_2}{0.5}, \left(0.25 \cdot {t\_6}^{1.5}\right) \cdot \left(t\_1 \cdot 0.125\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(t\_5, \pi \cdot 0.5, \left(-0.25 \cdot t\_6\right) \cdot t\_4\right)\right)}{\pi}, f, \mathsf{fma}\left(t\_2, t\_0, t\_2 \cdot t\_0\right)\right), f, \frac{4}{{\pi}^{0.5}} \cdot \frac{\mathsf{fma}\left(\log f, -1, \log \left({\left(\pi \cdot 0.5\right)}^{-1} \cdot 2\right)\right)}{{\pi}^{0.5}}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ (* PI 0.5) PI))
        (t_1 (* (* PI PI) PI))
        (t_2 (* (/ PI (* PI 0.5)) 0.0))
        (t_3 (* t_1 0.005208333333333333))
        (t_4 (pow (* PI 0.5) 2.0))
        (t_5
         (-
          (* (/ (* PI PI) PI) 0.125)
          (/ (fma t_1 0.005208333333333333 t_3) t_4)))
        (t_6 (pow t_2 2.0)))
   (*
    -1.0
    (fma
     (fma
      (/
       (fma
        0.6666666666666666
        (*
         (fma
          (* -1.5 t_2)
          (* t_5 t_4)
          (fma
           (* 3.0 (* PI 0.5))
           (- (/ (* t_1 0.0) (* PI 0.5)) (* (* -1.0 (/ t_3 PI)) (/ t_2 0.5)))
           (* (* 0.25 (pow t_6 1.5)) (* t_1 0.125))))
         f)
        (* 2.0 (fma t_5 (* PI 0.5) (* (* -0.25 t_6) t_4))))
       PI)
      f
      (fma t_2 t_0 (* t_2 t_0)))
     f
     (*
      (/ 4.0 (pow PI 0.5))
      (/
       (fma (log f) -1.0 (log (* (pow (* PI 0.5) -1.0) 2.0)))
       (pow PI 0.5)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) * 0.5) / ((double) M_PI);
	double t_1 = (((double) M_PI) * ((double) M_PI)) * ((double) M_PI);
	double t_2 = (((double) M_PI) / (((double) M_PI) * 0.5)) * 0.0;
	double t_3 = t_1 * 0.005208333333333333;
	double t_4 = pow((((double) M_PI) * 0.5), 2.0);
	double t_5 = (((((double) M_PI) * ((double) M_PI)) / ((double) M_PI)) * 0.125) - (fma(t_1, 0.005208333333333333, t_3) / t_4);
	double t_6 = pow(t_2, 2.0);
	return -1.0 * fma(fma((fma(0.6666666666666666, (fma((-1.5 * t_2), (t_5 * t_4), fma((3.0 * (((double) M_PI) * 0.5)), (((t_1 * 0.0) / (((double) M_PI) * 0.5)) - ((-1.0 * (t_3 / ((double) M_PI))) * (t_2 / 0.5))), ((0.25 * pow(t_6, 1.5)) * (t_1 * 0.125)))) * f), (2.0 * fma(t_5, (((double) M_PI) * 0.5), ((-0.25 * t_6) * t_4)))) / ((double) M_PI)), f, fma(t_2, t_0, (t_2 * t_0))), f, ((4.0 / pow(((double) M_PI), 0.5)) * (fma(log(f), -1.0, log((pow((((double) M_PI) * 0.5), -1.0) * 2.0))) / pow(((double) M_PI), 0.5))));
}

Julia

function code(f)
	t_0 = Float64(Float64(pi * 0.5) / pi)
	t_1 = Float64(Float64(pi * pi) * pi)
	t_2 = Float64(Float64(pi / Float64(pi * 0.5)) * 0.0)
	t_3 = Float64(t_1 * 0.005208333333333333)
	t_4 = Float64(pi * 0.5) ^ 2.0
	t_5 = Float64(Float64(Float64(Float64(pi * pi) / pi) * 0.125) - Float64(fma(t_1, 0.005208333333333333, t_3) / t_4))
	t_6 = t_2 ^ 2.0
	return Float64(-1.0 * fma(fma(Float64(fma(0.6666666666666666, Float64(fma(Float64(-1.5 * t_2), Float64(t_5 * t_4), fma(Float64(3.0 * Float64(pi * 0.5)), Float64(Float64(Float64(t_1 * 0.0) / Float64(pi * 0.5)) - Float64(Float64(-1.0 * Float64(t_3 / pi)) * Float64(t_2 / 0.5))), Float64(Float64(0.25 * (t_6 ^ 1.5)) * Float64(t_1 * 0.125)))) * f), Float64(2.0 * fma(t_5, Float64(pi * 0.5), Float64(Float64(-0.25 * t_6) * t_4)))) / pi), f, fma(t_2, t_0, Float64(t_2 * t_0))), f, Float64(Float64(4.0 / (pi ^ 0.5)) * Float64(fma(log(f), -1.0, log(Float64((Float64(pi * 0.5) ^ -1.0) * 2.0))) / (pi ^ 0.5)))))
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi * 0.5), $MachinePrecision] / Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * 0.005208333333333333), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(Pi * 0.5), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(Pi * Pi), $MachinePrecision] / Pi), $MachinePrecision] * 0.125), $MachinePrecision] - N[(N[(t$95$1 * 0.005208333333333333 + t$95$3), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[t$95$2, 2.0], $MachinePrecision]}, N[(-1.0 * N[(N[(N[(N[(0.6666666666666666 * N[(N[(N[(-1.5 * t$95$2), $MachinePrecision] * N[(t$95$5 * t$95$4), $MachinePrecision] + N[(N[(3.0 * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$1 * 0.0), $MachinePrecision] / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 * N[(t$95$3 / Pi), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.25 * N[Power[t$95$6, 1.5], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * f), $MachinePrecision] + N[(2.0 * N[(t$95$5 * N[(Pi * 0.5), $MachinePrecision] + N[(N[(-0.25 * t$95$6), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * f + N[(t$95$2 * t$95$0 + N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * f + N[(N[(4.0 / N[Power[Pi, 0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Log[f], $MachinePrecision] * -1.0 + N[Log[N[(N[Power[N[(Pi * 0.5), $MachinePrecision], -1.0], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Power[Pi, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 8.1%

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

3. Taylor expanded in f around 0

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

5. Applied rewrites 96.5%

   \[\leadsto -\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, \mathsf{fma}\left(-1.5 \cdot \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot 0.5\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot 0.5} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot 0.5} \cdot 0}{0.5}, \left(0.25 \cdot {\left({\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right)}^{1.5}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.125\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \pi \cdot 0.5, \left(-0.25 \cdot {\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0, \frac{\pi \cdot 0.5}{\pi}, \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right) \cdot \frac{\pi \cdot 0.5}{\pi}\right)\right), f, \frac{4}{{\pi}^{0.5}} \cdot \frac{\mathsf{fma}\left(\log f, -1, \log \left({\left(\pi \cdot 0.5\right)}^{-1} \cdot 2\right)\right)}{{\pi}^{0.5}}\right)} \]

6. Final simplification 96.5%

   \[\leadsto -1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, \mathsf{fma}\left(-1.5 \cdot \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot 0.5\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot 0.5} - \left(-1 \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333}{\pi}\right) \cdot \frac{\frac{\pi}{\pi \cdot 0.5} \cdot 0}{0.5}, \left(0.25 \cdot {\left({\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right)}^{1.5}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.125\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \pi \cdot 0.5, \left(-0.25 \cdot {\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0, \frac{\pi \cdot 0.5}{\pi}, \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right) \cdot \frac{\pi \cdot 0.5}{\pi}\right)\right), f, \frac{4}{{\pi}^{0.5}} \cdot \frac{\mathsf{fma}\left(\log f, -1, \log \left({\left(\pi \cdot 0.5\right)}^{-1} \cdot 2\right)\right)}{{\pi}^{0.5}}\right) \]
7. Add Preprocessing

Alternative 7: 96.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi \cdot 0.5}{\pi}\\ t_1 := \left(\pi \cdot \pi\right) \cdot \pi\\ t_2 := \frac{\pi}{\pi \cdot 0.5} \cdot 0\\ t_3 := t\_1 \cdot 0.005208333333333333\\ t_4 := {\left(\pi \cdot 0.5\right)}^{2}\\ t_5 := \frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(t\_1, 0.005208333333333333, t\_3\right)}{t\_4}\\ t_6 := {t\_2}^{2}\\ -1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, \mathsf{fma}\left(-1.5 \cdot t\_2, t\_5 \cdot t\_4, \mathsf{fma}\left(3 \cdot \left(\pi \cdot 0.5\right), \frac{t\_1 \cdot 0}{\pi \cdot 0.5} - \left(-1 \cdot \frac{t\_3}{\pi}\right) \cdot \frac{t\_2}{0.5}, \left(0.25 \cdot {t\_6}^{1.5}\right) \cdot \left(t\_1 \cdot 0.125\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(t\_5, \pi \cdot 0.5, \left(-0.25 \cdot t\_6\right) \cdot t\_4\right)\right)}{\pi}, f, \mathsf{fma}\left(t\_2, t\_0, t\_2 \cdot t\_0\right)\right), f, \frac{4}{{\pi}^{0.5}} \cdot \left({\left({\pi}^{-1}\right)}^{0.5} \cdot \mathsf{fma}\left(\log f, -1, \log 4 - \log \pi\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ (* PI 0.5) PI))
        (t_1 (* (* PI PI) PI))
        (t_2 (* (/ PI (* PI 0.5)) 0.0))
        (t_3 (* t_1 0.005208333333333333))
        (t_4 (pow (* PI 0.5) 2.0))
        (t_5
         (-
          (* (/ (* PI PI) PI) 0.125)
          (/ (fma t_1 0.005208333333333333 t_3) t_4)))
        (t_6 (pow t_2 2.0)))
   (*
    -1.0
    (fma
     (fma
      (/
       (fma
        0.6666666666666666
        (*
         (fma
          (* -1.5 t_2)
          (* t_5 t_4)
          (fma
           (* 3.0 (* PI 0.5))
           (- (/ (* t_1 0.0) (* PI 0.5)) (* (* -1.0 (/ t_3 PI)) (/ t_2 0.5)))
           (* (* 0.25 (pow t_6 1.5)) (* t_1 0.125))))
         f)
        (* 2.0 (fma t_5 (* PI 0.5) (* (* -0.25 t_6) t_4))))
       PI)
      f
      (fma t_2 t_0 (* t_2 t_0)))
     f
     (*
      (/ 4.0 (pow PI 0.5))
      (*
       (pow (pow PI -1.0) 0.5)
       (fma (log f) -1.0 (- (log 4.0) (log PI)))))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) * 0.5) / ((double) M_PI);
	double t_1 = (((double) M_PI) * ((double) M_PI)) * ((double) M_PI);
	double t_2 = (((double) M_PI) / (((double) M_PI) * 0.5)) * 0.0;
	double t_3 = t_1 * 0.005208333333333333;
	double t_4 = pow((((double) M_PI) * 0.5), 2.0);
	double t_5 = (((((double) M_PI) * ((double) M_PI)) / ((double) M_PI)) * 0.125) - (fma(t_1, 0.005208333333333333, t_3) / t_4);
	double t_6 = pow(t_2, 2.0);
	return -1.0 * fma(fma((fma(0.6666666666666666, (fma((-1.5 * t_2), (t_5 * t_4), fma((3.0 * (((double) M_PI) * 0.5)), (((t_1 * 0.0) / (((double) M_PI) * 0.5)) - ((-1.0 * (t_3 / ((double) M_PI))) * (t_2 / 0.5))), ((0.25 * pow(t_6, 1.5)) * (t_1 * 0.125)))) * f), (2.0 * fma(t_5, (((double) M_PI) * 0.5), ((-0.25 * t_6) * t_4)))) / ((double) M_PI)), f, fma(t_2, t_0, (t_2 * t_0))), f, ((4.0 / pow(((double) M_PI), 0.5)) * (pow(pow(((double) M_PI), -1.0), 0.5) * fma(log(f), -1.0, (log(4.0) - log(((double) M_PI)))))));
}

Julia

function code(f)
	t_0 = Float64(Float64(pi * 0.5) / pi)
	t_1 = Float64(Float64(pi * pi) * pi)
	t_2 = Float64(Float64(pi / Float64(pi * 0.5)) * 0.0)
	t_3 = Float64(t_1 * 0.005208333333333333)
	t_4 = Float64(pi * 0.5) ^ 2.0
	t_5 = Float64(Float64(Float64(Float64(pi * pi) / pi) * 0.125) - Float64(fma(t_1, 0.005208333333333333, t_3) / t_4))
	t_6 = t_2 ^ 2.0
	return Float64(-1.0 * fma(fma(Float64(fma(0.6666666666666666, Float64(fma(Float64(-1.5 * t_2), Float64(t_5 * t_4), fma(Float64(3.0 * Float64(pi * 0.5)), Float64(Float64(Float64(t_1 * 0.0) / Float64(pi * 0.5)) - Float64(Float64(-1.0 * Float64(t_3 / pi)) * Float64(t_2 / 0.5))), Float64(Float64(0.25 * (t_6 ^ 1.5)) * Float64(t_1 * 0.125)))) * f), Float64(2.0 * fma(t_5, Float64(pi * 0.5), Float64(Float64(-0.25 * t_6) * t_4)))) / pi), f, fma(t_2, t_0, Float64(t_2 * t_0))), f, Float64(Float64(4.0 / (pi ^ 0.5)) * Float64(((pi ^ -1.0) ^ 0.5) * fma(log(f), -1.0, Float64(log(4.0) - log(pi)))))))
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi * 0.5), $MachinePrecision] / Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * 0.005208333333333333), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(Pi * 0.5), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(Pi * Pi), $MachinePrecision] / Pi), $MachinePrecision] * 0.125), $MachinePrecision] - N[(N[(t$95$1 * 0.005208333333333333 + t$95$3), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[t$95$2, 2.0], $MachinePrecision]}, N[(-1.0 * N[(N[(N[(N[(0.6666666666666666 * N[(N[(N[(-1.5 * t$95$2), $MachinePrecision] * N[(t$95$5 * t$95$4), $MachinePrecision] + N[(N[(3.0 * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$1 * 0.0), $MachinePrecision] / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 * N[(t$95$3 / Pi), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.25 * N[Power[t$95$6, 1.5], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * f), $MachinePrecision] + N[(2.0 * N[(t$95$5 * N[(Pi * 0.5), $MachinePrecision] + N[(N[(-0.25 * t$95$6), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * f + N[(t$95$2 * t$95$0 + N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * f + N[(N[(4.0 / N[Power[Pi, 0.5], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Power[Pi, -1.0], $MachinePrecision], 0.5], $MachinePrecision] * N[(N[Log[f], $MachinePrecision] * -1.0 + N[(N[Log[4.0], $MachinePrecision] - N[Log[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 8.1%

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

3. Taylor expanded in f around 0

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

5. Applied rewrites 96.5%

   \[\leadsto -\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, \mathsf{fma}\left(-1.5 \cdot \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot 0.5\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot 0.5} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot 0.5} \cdot 0}{0.5}, \left(0.25 \cdot {\left({\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right)}^{1.5}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.125\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \pi \cdot 0.5, \left(-0.25 \cdot {\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0, \frac{\pi \cdot 0.5}{\pi}, \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right) \cdot \frac{\pi \cdot 0.5}{\pi}\right)\right), f, \frac{4}{{\pi}^{0.5}} \cdot \frac{\mathsf{fma}\left(\log f, -1, \log \left({\left(\pi \cdot 0.5\right)}^{-1} \cdot 2\right)\right)}{{\pi}^{0.5}}\right)} \]

6. Taylor expanded in f around 0

   \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f\right)\right)\right) \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f\right)\right)\right) \]

   2. pow1/2 N/A

      \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \left({\left(\frac{1}{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}} \cdot \left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f\right)\right)\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \left({\left(\frac{1}{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}} \cdot \left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f\right)\right)\right) \]

   4. inv-pow N/A

      \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \left({\left({\mathsf{PI}\left(\right)}^{-1}\right)}^{\frac{1}{2}} \cdot \left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f\right)\right)\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \left({\left({\mathsf{PI}\left(\right)}^{-1}\right)}^{\frac{1}{2}} \cdot \left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f\right)\right)\right) \]

   6. lift-PI.f64 N/A

      \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \left({\left({\pi}^{-1}\right)}^{\frac{1}{2}} \cdot \left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \left({\left({\pi}^{-1}\right)}^{\frac{1}{2}} \cdot \left(-1 \cdot \log f + \log \left(\frac{4}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \left({\left({\pi}^{-1}\right)}^{\frac{1}{2}} \cdot \left(\log f \cdot -1 + \log \left(\frac{4}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \left({\left({\pi}^{-1}\right)}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\log f, -1, \log \left(\frac{4}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

   10. lift-log.f64 N/A

       \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \left({\left({\pi}^{-1}\right)}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\log f, -1, \log \left(\frac{4}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

   11. log-div N/A

       \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \left({\left({\pi}^{-1}\right)}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\log f, -1, \log 4 - \log \mathsf{PI}\left(\right)\right)\right)\right) \]

   12. lower--.f64 N/A

       \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \left({\left({\pi}^{-1}\right)}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\log f, -1, \log 4 - \log \mathsf{PI}\left(\right)\right)\right)\right) \]

   13. lower-log.f64 N/A

       \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \left({\left({\pi}^{-1}\right)}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\log f, -1, \log 4 - \log \mathsf{PI}\left(\right)\right)\right)\right) \]

   14. lower-log.f64 N/A

       \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3}, \mathsf{fma}\left(\frac{-3}{2} \cdot \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot \frac{1}{2}\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot \frac{1}{2}} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0}{\frac{1}{2}}, \left(\frac{1}{4} \cdot {\left({\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right)}^{\frac{3}{2}}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{8}\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot \frac{1}{8} - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{1}{192}, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{192}\right)}{{\left(\pi \cdot \frac{1}{2}\right)}^{2}}, \pi \cdot \frac{1}{2}, \left(\frac{-1}{4} \cdot {\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot \frac{1}{2}\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0, \frac{\pi \cdot \frac{1}{2}}{\pi}, \left(\frac{\pi}{\pi \cdot \frac{1}{2}} \cdot 0\right) \cdot \frac{\pi \cdot \frac{1}{2}}{\pi}\right)\right), f, \frac{4}{{\pi}^{\frac{1}{2}}} \cdot \left({\left({\pi}^{-1}\right)}^{\frac{1}{2}} \cdot \mathsf{fma}\left(\log f, -1, \log 4 - \log \mathsf{PI}\left(\right)\right)\right)\right) \]

   15. lift-PI.f64 96.5

       \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, \mathsf{fma}\left(-1.5 \cdot \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot 0.5\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot 0.5} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot 0.5} \cdot 0}{0.5}, \left(0.25 \cdot {\left({\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right)}^{1.5}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.125\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \pi \cdot 0.5, \left(-0.25 \cdot {\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0, \frac{\pi \cdot 0.5}{\pi}, \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right) \cdot \frac{\pi \cdot 0.5}{\pi}\right)\right), f, \frac{4}{{\pi}^{0.5}} \cdot \left({\left({\pi}^{-1}\right)}^{0.5} \cdot \mathsf{fma}\left(\log f, -1, \log 4 - \log \pi\right)\right)\right) \]

8. Applied rewrites 96.5%

   \[\leadsto -\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, \mathsf{fma}\left(-1.5 \cdot \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot 0.5\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot 0.5} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot 0.5} \cdot 0}{0.5}, \left(0.25 \cdot {\left({\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right)}^{1.5}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.125\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \pi \cdot 0.5, \left(-0.25 \cdot {\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0, \frac{\pi \cdot 0.5}{\pi}, \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right) \cdot \frac{\pi \cdot 0.5}{\pi}\right)\right), f, \frac{4}{{\pi}^{0.5}} \cdot \left({\left({\pi}^{-1}\right)}^{0.5} \cdot \mathsf{fma}\left(\log f, -1, \log 4 - \log \pi\right)\right)\right) \]

9. Final simplification 96.5%

   \[\leadsto -1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, \mathsf{fma}\left(-1.5 \cdot \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot 0.5\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot 0.5} - \left(-1 \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333}{\pi}\right) \cdot \frac{\frac{\pi}{\pi \cdot 0.5} \cdot 0}{0.5}, \left(0.25 \cdot {\left({\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right)}^{1.5}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.125\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \pi \cdot 0.5, \left(-0.25 \cdot {\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0, \frac{\pi \cdot 0.5}{\pi}, \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right) \cdot \frac{\pi \cdot 0.5}{\pi}\right)\right), f, \frac{4}{{\pi}^{0.5}} \cdot \left({\left({\pi}^{-1}\right)}^{0.5} \cdot \mathsf{fma}\left(\log f, -1, \log 4 - \log \pi\right)\right)\right) \]
10. Add Preprocessing

Alternative 8: 96.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot \pi\right) \cdot \pi\\ t_1 := \left(\pi \cdot \pi\right) \cdot 0.25\\ t_2 := \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{t\_0 \cdot 0.010416666666666666}{t\_1}\\ t_3 := \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(t\_1 \cdot t\_2, 0, \mathsf{fma}\left(\frac{t\_0}{\pi} \cdot 0 - \left(\left(-1 \cdot t\_0\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, t\_2, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\\ t_4 := \frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\\ -1 \cdot \frac{{t\_3}^{3} + {t\_4}^{3}}{t\_3 \cdot t\_3 + \left(t\_4 \cdot t\_4 - t\_3 \cdot t\_4\right)} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (* PI PI) PI))
        (t_1 (* (* PI PI) 0.25))
        (t_2
         (- (/ (* (* PI PI) 0.125) PI) (/ (* t_0 0.010416666666666666) t_1)))
        (t_3
         (*
          (fma
           (/
            (fma
             (*
              0.6666666666666666
              (fma
               (* t_1 t_2)
               0.0
               (fma
                (-
                 (* (/ t_0 PI) 0.0)
                 (* (* (* -1.0 t_0) (/ 0.005208333333333333 PI)) 0.0))
                (* (* 3.0 PI) 0.5)
                0.0)))
             f
             (* (fma (* 0.5 PI) t_2 0.0) 2.0))
            PI)
           f
           (/ 0.0 PI))
          f))
        (t_4
         (*
          (/ (fma (log f) -1.0 (log (/ 2.0 (* 0.5 PI)))) (pow PI 0.5))
          (/ 4.0 (pow PI 0.5)))))
   (*
    -1.0
    (/
     (+ (pow t_3 3.0) (pow t_4 3.0))
     (+ (* t_3 t_3) (- (* t_4 t_4) (* t_3 t_4)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) * ((double) M_PI)) * ((double) M_PI);
	double t_1 = (((double) M_PI) * ((double) M_PI)) * 0.25;
	double t_2 = (((((double) M_PI) * ((double) M_PI)) * 0.125) / ((double) M_PI)) - ((t_0 * 0.010416666666666666) / t_1);
	double t_3 = fma((fma((0.6666666666666666 * fma((t_1 * t_2), 0.0, fma((((t_0 / ((double) M_PI)) * 0.0) - (((-1.0 * t_0) * (0.005208333333333333 / ((double) M_PI))) * 0.0)), ((3.0 * ((double) M_PI)) * 0.5), 0.0))), f, (fma((0.5 * ((double) M_PI)), t_2, 0.0) * 2.0)) / ((double) M_PI)), f, (0.0 / ((double) M_PI))) * f;
	double t_4 = (fma(log(f), -1.0, log((2.0 / (0.5 * ((double) M_PI))))) / pow(((double) M_PI), 0.5)) * (4.0 / pow(((double) M_PI), 0.5));
	return -1.0 * ((pow(t_3, 3.0) + pow(t_4, 3.0)) / ((t_3 * t_3) + ((t_4 * t_4) - (t_3 * t_4))));
}

Julia

function code(f)
	t_0 = Float64(Float64(pi * pi) * pi)
	t_1 = Float64(Float64(pi * pi) * 0.25)
	t_2 = Float64(Float64(Float64(Float64(pi * pi) * 0.125) / pi) - Float64(Float64(t_0 * 0.010416666666666666) / t_1))
	t_3 = Float64(fma(Float64(fma(Float64(0.6666666666666666 * fma(Float64(t_1 * t_2), 0.0, fma(Float64(Float64(Float64(t_0 / pi) * 0.0) - Float64(Float64(Float64(-1.0 * t_0) * Float64(0.005208333333333333 / pi)) * 0.0)), Float64(Float64(3.0 * pi) * 0.5), 0.0))), f, Float64(fma(Float64(0.5 * pi), t_2, 0.0) * 2.0)) / pi), f, Float64(0.0 / pi)) * f)
	t_4 = Float64(Float64(fma(log(f), -1.0, log(Float64(2.0 / Float64(0.5 * pi)))) / (pi ^ 0.5)) * Float64(4.0 / (pi ^ 0.5)))
	return Float64(-1.0 * Float64(Float64((t_3 ^ 3.0) + (t_4 ^ 3.0)) / Float64(Float64(t_3 * t_3) + Float64(Float64(t_4 * t_4) - Float64(t_3 * t_4)))))
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.125), $MachinePrecision] / Pi), $MachinePrecision] - N[(N[(t$95$0 * 0.010416666666666666), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(0.6666666666666666 * N[(N[(t$95$1 * t$95$2), $MachinePrecision] * 0.0 + N[(N[(N[(N[(t$95$0 / Pi), $MachinePrecision] * 0.0), $MachinePrecision] - N[(N[(N[(-1.0 * t$95$0), $MachinePrecision] * N[(0.005208333333333333 / Pi), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] * N[(N[(3.0 * Pi), $MachinePrecision] * 0.5), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * f + N[(N[(N[(0.5 * Pi), $MachinePrecision] * t$95$2 + 0.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * f + N[(0.0 / Pi), $MachinePrecision]), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Log[f], $MachinePrecision] * -1.0 + N[Log[N[(2.0 / N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Power[Pi, 0.5], $MachinePrecision]), $MachinePrecision] * N[(4.0 / N[Power[Pi, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(-1.0 * N[(N[(N[Power[t$95$3, 3.0], $MachinePrecision] + N[Power[t$95$4, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$3 * t$95$3), $MachinePrecision] + N[(N[(t$95$4 * t$95$4), $MachinePrecision] - N[(t$95$3 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 8.1%

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

3. Taylor expanded in f around 0

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

5. Applied rewrites 96.5%

   \[\leadsto -\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, \mathsf{fma}\left(-1.5 \cdot \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot 0.5\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot 0.5} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot 0.5} \cdot 0}{0.5}, \left(0.25 \cdot {\left({\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right)}^{1.5}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.125\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \pi \cdot 0.5, \left(-0.25 \cdot {\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0, \frac{\pi \cdot 0.5}{\pi}, \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right) \cdot \frac{\pi \cdot 0.5}{\pi}\right)\right), f, \frac{4}{{\pi}^{0.5}} \cdot \frac{\mathsf{fma}\left(\log f, -1, \log \left({\left(\pi \cdot 0.5\right)}^{-1} \cdot 2\right)\right)}{{\pi}^{0.5}}\right)} \]

7. Applied rewrites 96.4%

   \[\leadsto -\frac{{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right)}^{3} + {\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right)}^{3}}{\color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) + \left(\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right)\right)}} \]

8. Final simplification 96.4%

   \[\leadsto -1 \cdot \frac{{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(-1 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right)}^{3} + {\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right)}^{3}}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(-1 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(-1 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) + \left(\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(-1 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right)\right)} \]
9. Add Preprocessing

Alternative 9: 95.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot \pi\right) \cdot \pi\\ t_1 := \frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\\ t_2 := \left(\pi \cdot \pi\right) \cdot 0.25\\ t_3 := \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{t\_0 \cdot 0.010416666666666666}{t\_2}\\ t_4 := \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(t\_2 \cdot t\_3, 0, \mathsf{fma}\left(\frac{t\_0}{\pi} \cdot 0 - \left(\left(-1 \cdot t\_0\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, t\_3, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\\ -1 \cdot \frac{64 \cdot \frac{{\left(\left(\log 4 - \log \pi\right) - \log f\right)}^{3}}{t\_0}}{t\_4 \cdot t\_4 + \left(t\_1 \cdot t\_1 - t\_4 \cdot t\_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (* PI PI) PI))
        (t_1
         (*
          (/ (fma (log f) -1.0 (log (/ 2.0 (* 0.5 PI)))) (pow PI 0.5))
          (/ 4.0 (pow PI 0.5))))
        (t_2 (* (* PI PI) 0.25))
        (t_3
         (- (/ (* (* PI PI) 0.125) PI) (/ (* t_0 0.010416666666666666) t_2)))
        (t_4
         (*
          (fma
           (/
            (fma
             (*
              0.6666666666666666
              (fma
               (* t_2 t_3)
               0.0
               (fma
                (-
                 (* (/ t_0 PI) 0.0)
                 (* (* (* -1.0 t_0) (/ 0.005208333333333333 PI)) 0.0))
                (* (* 3.0 PI) 0.5)
                0.0)))
             f
             (* (fma (* 0.5 PI) t_3 0.0) 2.0))
            PI)
           f
           (/ 0.0 PI))
          f)))
   (*
    -1.0
    (/
     (* 64.0 (/ (pow (- (- (log 4.0) (log PI)) (log f)) 3.0) t_0))
     (+ (* t_4 t_4) (- (* t_1 t_1) (* t_4 t_1)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) * ((double) M_PI)) * ((double) M_PI);
	double t_1 = (fma(log(f), -1.0, log((2.0 / (0.5 * ((double) M_PI))))) / pow(((double) M_PI), 0.5)) * (4.0 / pow(((double) M_PI), 0.5));
	double t_2 = (((double) M_PI) * ((double) M_PI)) * 0.25;
	double t_3 = (((((double) M_PI) * ((double) M_PI)) * 0.125) / ((double) M_PI)) - ((t_0 * 0.010416666666666666) / t_2);
	double t_4 = fma((fma((0.6666666666666666 * fma((t_2 * t_3), 0.0, fma((((t_0 / ((double) M_PI)) * 0.0) - (((-1.0 * t_0) * (0.005208333333333333 / ((double) M_PI))) * 0.0)), ((3.0 * ((double) M_PI)) * 0.5), 0.0))), f, (fma((0.5 * ((double) M_PI)), t_3, 0.0) * 2.0)) / ((double) M_PI)), f, (0.0 / ((double) M_PI))) * f;
	return -1.0 * ((64.0 * (pow(((log(4.0) - log(((double) M_PI))) - log(f)), 3.0) / t_0)) / ((t_4 * t_4) + ((t_1 * t_1) - (t_4 * t_1))));
}

Julia

function code(f)
	t_0 = Float64(Float64(pi * pi) * pi)
	t_1 = Float64(Float64(fma(log(f), -1.0, log(Float64(2.0 / Float64(0.5 * pi)))) / (pi ^ 0.5)) * Float64(4.0 / (pi ^ 0.5)))
	t_2 = Float64(Float64(pi * pi) * 0.25)
	t_3 = Float64(Float64(Float64(Float64(pi * pi) * 0.125) / pi) - Float64(Float64(t_0 * 0.010416666666666666) / t_2))
	t_4 = Float64(fma(Float64(fma(Float64(0.6666666666666666 * fma(Float64(t_2 * t_3), 0.0, fma(Float64(Float64(Float64(t_0 / pi) * 0.0) - Float64(Float64(Float64(-1.0 * t_0) * Float64(0.005208333333333333 / pi)) * 0.0)), Float64(Float64(3.0 * pi) * 0.5), 0.0))), f, Float64(fma(Float64(0.5 * pi), t_3, 0.0) * 2.0)) / pi), f, Float64(0.0 / pi)) * f)
	return Float64(-1.0 * Float64(Float64(64.0 * Float64((Float64(Float64(log(4.0) - log(pi)) - log(f)) ^ 3.0) / t_0)) / Float64(Float64(t_4 * t_4) + Float64(Float64(t_1 * t_1) - Float64(t_4 * t_1)))))
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Log[f], $MachinePrecision] * -1.0 + N[Log[N[(2.0 / N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Power[Pi, 0.5], $MachinePrecision]), $MachinePrecision] * N[(4.0 / N[Power[Pi, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.125), $MachinePrecision] / Pi), $MachinePrecision] - N[(N[(t$95$0 * 0.010416666666666666), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(0.6666666666666666 * N[(N[(t$95$2 * t$95$3), $MachinePrecision] * 0.0 + N[(N[(N[(N[(t$95$0 / Pi), $MachinePrecision] * 0.0), $MachinePrecision] - N[(N[(N[(-1.0 * t$95$0), $MachinePrecision] * N[(0.005208333333333333 / Pi), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] * N[(N[(3.0 * Pi), $MachinePrecision] * 0.5), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * f + N[(N[(N[(0.5 * Pi), $MachinePrecision] * t$95$3 + 0.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * f + N[(0.0 / Pi), $MachinePrecision]), $MachinePrecision] * f), $MachinePrecision]}, N[(-1.0 * N[(N[(64.0 * N[(N[Power[N[(N[(N[Log[4.0], $MachinePrecision] - N[Log[Pi], $MachinePrecision]), $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$4 * t$95$4), $MachinePrecision] + N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(t$95$4 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 8.1%

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

3. Taylor expanded in f around 0

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

5. Applied rewrites 96.5%

   \[\leadsto -\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, \mathsf{fma}\left(-1.5 \cdot \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right), \left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}, \mathsf{fma}\left(3 \cdot \left(\pi \cdot 0.5\right), \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0}{\pi \cdot 0.5} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333}{\pi} \cdot \frac{\frac{\pi}{\pi \cdot 0.5} \cdot 0}{0.5}, \left(0.25 \cdot {\left({\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right)}^{1.5}\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.125\right)\right)\right) \cdot f, 2 \cdot \mathsf{fma}\left(\frac{\pi \cdot \pi}{\pi} \cdot 0.125 - \frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, 0.005208333333333333, \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.005208333333333333\right)}{{\left(\pi \cdot 0.5\right)}^{2}}, \pi \cdot 0.5, \left(-0.25 \cdot {\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right)}^{2}\right) \cdot {\left(\pi \cdot 0.5\right)}^{2}\right)\right)}{\pi}, f, \mathsf{fma}\left(\frac{\pi}{\pi \cdot 0.5} \cdot 0, \frac{\pi \cdot 0.5}{\pi}, \left(\frac{\pi}{\pi \cdot 0.5} \cdot 0\right) \cdot \frac{\pi \cdot 0.5}{\pi}\right)\right), f, \frac{4}{{\pi}^{0.5}} \cdot \frac{\mathsf{fma}\left(\log f, -1, \log \left({\left(\pi \cdot 0.5\right)}^{-1} \cdot 2\right)\right)}{{\pi}^{0.5}}\right)} \]

7. Applied rewrites 96.4%

   \[\leadsto -\frac{{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right)}^{3} + {\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right)}^{3}}{\color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) + \left(\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right)\right)}} \]

8. Taylor expanded in f around 0

   \[\leadsto -\frac{64 \cdot \frac{{\left(\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \log f\right)}^{3}}{{\mathsf{PI}\left(\right)}^{3}}}{\color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right)} + \left(\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right)\right)} \]
9. Step-by-step derivation

10. Applied rewrites 95.8%

    \[\leadsto -\frac{64 \cdot {\left(\frac{\left(\log 4 - \log \pi\right) + -1 \cdot \log f}{\pi}\right)}^{3}}{\color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right)} + \left(\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right)\right)} \]
11. Step-by-step derivation

    1. lift-pow.f64 N/A

       \[\leadsto -\frac{64 \cdot {\left(\frac{\left(\log 4 - \log \pi\right) + -1 \cdot \log f}{\pi}\right)}^{3}}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot \color{blue}{f}\right) + \left(\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right)\right)} \]

    2. lift-PI.f64 N/A

       \[\leadsto -\frac{64 \cdot {\left(\frac{\left(\log 4 - \log \pi\right) + -1 \cdot \log f}{\mathsf{PI}\left(\right)}\right)}^{3}}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) + \left(\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right)\right)} \]

    3. lift-/.f64 N/A

       \[\leadsto -\frac{64 \cdot {\left(\frac{\left(\log 4 - \log \pi\right) + -1 \cdot \log f}{\mathsf{PI}\left(\right)}\right)}^{3}}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) + \left(\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right)\right)} \]

    4. lift-+.f64 N/A

       \[\leadsto -\frac{64 \cdot {\left(\frac{\left(\log 4 - \log \pi\right) + -1 \cdot \log f}{\mathsf{PI}\left(\right)}\right)}^{3}}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) + \left(\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right)\right)} \]

    5. lift--.f64 N/A

       \[\leadsto -\frac{64 \cdot {\left(\frac{\left(\log 4 - \log \pi\right) + -1 \cdot \log f}{\mathsf{PI}\left(\right)}\right)}^{3}}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) + \left(\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right)\right)} \]

    6. lift-log.f64 N/A

       \[\leadsto -\frac{64 \cdot {\left(\frac{\left(\log 4 - \log \pi\right) + -1 \cdot \log f}{\mathsf{PI}\left(\right)}\right)}^{3}}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) + \left(\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right)\right)} \]

    7. lift-PI.f64 N/A

       \[\leadsto -\frac{64 \cdot {\left(\frac{\left(\log 4 - \log \mathsf{PI}\left(\right)\right) + -1 \cdot \log f}{\mathsf{PI}\left(\right)}\right)}^{3}}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) + \left(\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right)\right)} \]

    8. lift-log.f64 N/A

       \[\leadsto -\frac{64 \cdot {\left(\frac{\left(\log 4 - \log \mathsf{PI}\left(\right)\right) + -1 \cdot \log f}{\mathsf{PI}\left(\right)}\right)}^{3}}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) + \left(\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right)\right)} \]

    9. lift-*.f64 N/A

       \[\leadsto -\frac{64 \cdot {\left(\frac{\left(\log 4 - \log \mathsf{PI}\left(\right)\right) + -1 \cdot \log f}{\mathsf{PI}\left(\right)}\right)}^{3}}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) + \left(\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right)\right)} \]

    10. lift-log.f64 N/A

        \[\leadsto -\frac{64 \cdot {\left(\frac{\left(\log 4 - \log \mathsf{PI}\left(\right)\right) + -1 \cdot \log f}{\mathsf{PI}\left(\right)}\right)}^{3}}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) + \left(\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right)\right)} \]

    11. cube-div N/A

        \[\leadsto -\frac{64 \cdot \frac{{\left(\left(\log 4 - \log \mathsf{PI}\left(\right)\right) + -1 \cdot \log f\right)}^{3}}{{\mathsf{PI}\left(\right)}^{3}}}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot \color{blue}{f}\right) + \left(\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{2}{3} \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \frac{1}{4}\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{\frac{1}{192}}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot \frac{1}{2}, 0\right)\right), f, \mathsf{fma}\left(\frac{1}{2} \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot \frac{1}{8}}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{1}{96}}{\left(\pi \cdot \pi\right) \cdot \frac{1}{4}}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{\frac{1}{2} \cdot \pi}\right)\right)}{{\pi}^{\frac{1}{2}}} \cdot \frac{4}{{\pi}^{\frac{1}{2}}}\right)\right)} \]

12. Applied rewrites 96.0%

    \[\leadsto -\frac{64 \cdot \frac{{\left(\left(\log 4 - \log \pi\right) + -1 \cdot \log f\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot \pi}}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot \color{blue}{f}\right) + \left(\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right)\right)} \]

13. Final simplification 96.0%

    \[\leadsto -1 \cdot \frac{64 \cdot \frac{{\left(\left(\log 4 - \log \pi\right) - \log f\right)}^{3}}{\left(\pi \cdot \pi\right) \cdot \pi}}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(-1 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(-1 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) + \left(\left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right) - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666 \cdot \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot 0.25\right) \cdot \left(\frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}\right), 0, \mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \pi}{\pi} \cdot 0 - \left(\left(-1 \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \frac{0.005208333333333333}{\pi}\right) \cdot 0, \left(3 \cdot \pi\right) \cdot 0.5, 0\right)\right), f, \mathsf{fma}\left(0.5 \cdot \pi, \frac{\left(\pi \cdot \pi\right) \cdot 0.125}{\pi} - \frac{\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot 0.010416666666666666}{\left(\pi \cdot \pi\right) \cdot 0.25}, 0\right) \cdot 2\right)}{\pi}, f, \frac{0}{\pi}\right) \cdot f\right) \cdot \left(\frac{\mathsf{fma}\left(\log f, -1, \log \left(\frac{2}{0.5 \cdot \pi}\right)\right)}{{\pi}^{0.5}} \cdot \frac{4}{{\pi}^{0.5}}\right)\right)} \]
14. Add Preprocessing

Alternative 10: 80.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := f \cdot \frac{\pi}{4}\\ \log \left({\left(\frac{\cosh t\_0}{\sinh t\_0}\right)}^{\left(-1 \cdot {\left(\frac{\pi}{4}\right)}^{-1}\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* f (/ PI 4.0))))
   (log (pow (/ (cosh t_0) (sinh t_0)) (* -1.0 (pow (/ PI 4.0) -1.0))))))

C

double code(double f) {
	double t_0 = f * (((double) M_PI) / 4.0);
	return log(pow((cosh(t_0) / sinh(t_0)), (-1.0 * pow((((double) M_PI) / 4.0), -1.0))));
}

Java

public static double code(double f) {
	double t_0 = f * (Math.PI / 4.0);
	return Math.log(Math.pow((Math.cosh(t_0) / Math.sinh(t_0)), (-1.0 * Math.pow((Math.PI / 4.0), -1.0))));
}

Python

def code(f):
	t_0 = f * (math.pi / 4.0)
	return math.log(math.pow((math.cosh(t_0) / math.sinh(t_0)), (-1.0 * math.pow((math.pi / 4.0), -1.0))))

Julia

function code(f)
	t_0 = Float64(f * Float64(pi / 4.0))
	return log((Float64(cosh(t_0) / sinh(t_0)) ^ Float64(-1.0 * (Float64(pi / 4.0) ^ -1.0))))
end

MATLAB

function tmp = code(f)
	t_0 = f * (pi / 4.0);
	tmp = log(((cosh(t_0) / sinh(t_0)) ^ (-1.0 * ((pi / 4.0) ^ -1.0))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(f * N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]}, N[Log[N[Power[N[(N[Cosh[t$95$0], $MachinePrecision] / N[Sinh[t$95$0], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[Power[N[(Pi / 4.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 8.1%

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

4. Applied rewrites 81.0%

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

5. Final simplification 81.0%

   \[\leadsto \log \left({\left(\frac{\cosh \left(f \cdot \frac{\pi}{4}\right)}{\sinh \left(f \cdot \frac{\pi}{4}\right)}\right)}^{\left(-1 \cdot {\left(\frac{\pi}{4}\right)}^{-1}\right)}\right) \]
6. Add Preprocessing

Alternative 11: 79.9% accurate, N/A× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* f (/ PI 4.0))) (t_1 (* (- (log PI) (log 4.0)) -1.0)))
   (log (pow (/ (cosh t_0) (sinh t_0)) (* -1.0 (+ (cosh t_1) (sinh t_1)))))))

C

double code(double f) {
	double t_0 = f * (((double) M_PI) / 4.0);
	double t_1 = (log(((double) M_PI)) - log(4.0)) * -1.0;
	return log(pow((cosh(t_0) / sinh(t_0)), (-1.0 * (cosh(t_1) + sinh(t_1)))));
}

Java

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

Python

def code(f):
	t_0 = f * (math.pi / 4.0)
	t_1 = (math.log(math.pi) - math.log(4.0)) * -1.0
	return math.log(math.pow((math.cosh(t_0) / math.sinh(t_0)), (-1.0 * (math.cosh(t_1) + math.sinh(t_1)))))

Julia

function code(f)
	t_0 = Float64(f * Float64(pi / 4.0))
	t_1 = Float64(Float64(log(pi) - log(4.0)) * -1.0)
	return log((Float64(cosh(t_0) / sinh(t_0)) ^ Float64(-1.0 * Float64(cosh(t_1) + sinh(t_1)))))
end

MATLAB

function tmp = code(f)
	t_0 = f * (pi / 4.0);
	t_1 = (log(pi) - log(4.0)) * -1.0;
	tmp = log(((cosh(t_0) / sinh(t_0)) ^ (-1.0 * (cosh(t_1) + sinh(t_1)))));
end

Wolfram

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

Derivation

1. Initial program 8.1%

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

4. Applied rewrites 81.0%

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

   1. lift-pow.f64 N/A

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

   2. pow-to-exp N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. lower-+.f64 N/A

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

   5. lower-cosh.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. log-div N/A

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

   10. lower--.f64 N/A

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

   11. lower-log.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-log.f64 N/A

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

   14. lower-sinh.f64 N/A

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

   15. lower-*.f64 N/A

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

6. Applied rewrites 80.7%

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

7. Final simplification 80.7%

   \[\leadsto \log \left({\left(\frac{\cosh \left(f \cdot \frac{\pi}{4}\right)}{\sinh \left(f \cdot \frac{\pi}{4}\right)}\right)}^{\left(-1 \cdot \left(\cosh \left(\left(\log \pi - \log 4\right) \cdot -1\right) + \sinh \left(\left(\log \pi - \log 4\right) \cdot -1\right)\right)\right)}\right) \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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)))))))))