Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I

Percentage Accurate: 94.2% → 97.1%

Time: 13.9s

Alternatives: 12

Speedup: 0.7×

Specification

Math

\[\begin{array}{l} \\ \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/
  x
  (+
   x
   (*
    y
    (exp
     (*
      2.0
      (-
       (/ (* z (sqrt (+ t a))) t)
       (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = x / (x + (y * exp((2.0d0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0)))))))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return x / (x + (y * Math.exp((2.0 * (((z * Math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
}

Python

def code(x, y, z, t, a, b, c):
	return x / (x + (y * math.exp((2.0 * (((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))))

Julia

function code(x, y, z, t, a, b, c)
	return Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) - Float64(Float64(b - c) * Float64(Float64(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0))))))))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/
  x
  (+
   x
   (*
    y
    (exp
     (*
      2.0
      (-
       (/ (* z (sqrt (+ t a))) t)
       (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = x / (x + (y * exp((2.0d0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0)))))))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return x / (x + (y * Math.exp((2.0 * (((z * Math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
}

Python

def code(x, y, z, t, a, b, c):
	return x / (x + (y * math.exp((2.0 * (((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))))

Julia

function code(x, y, z, t, a, b, c)
	return Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) - Float64(Float64(b - c) * Float64(Float64(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0))))))))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{\left(a \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (-
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))
   (if (<= t_1 INFINITY)
     (/ x (+ x (* y (exp (* 2.0 t_1)))))
     (/ x (fma (exp (* (* a (- b c)) -2.0)) y x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = x / (x + (y * exp((2.0 * t_1))));
	} else {
		tmp = x / fma(exp(((a * (b - c)) * -2.0)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) - Float64(Float64(b - c) * Float64(Float64(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0)))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * t_1)))));
	else
		tmp = Float64(x / fma(exp(Float64(Float64(a * Float64(b - c)) * -2.0)), y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[Exp[N[(N[(a * N[(b - c), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < +inf.0

   1. Initial program 99.6%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   if +inf.0 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))

   1. Initial program 0.0%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} + x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} \cdot y} + x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)}} \]

      4. exp-prod N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\color{blue}{\left(e^{-2}\right)}}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)} \cdot \left(b - c\right)\right)}, y, x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3}}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      14. lower--.f64 59.6

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \color{blue}{\left(b - c\right)}\right)}, y, x\right)} \]

   5. Applied rewrites 59.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.6%

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 51.6%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]

   11. Taylor expanded in a around inf

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(a \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 59.6%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(a \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)} \leq 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{\left(0.8333333333333334 \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (exp
       (*
        2.0
        (-
         (/ (* z (sqrt (+ t a))) t)
         (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))
      0.0)
   1.0
   (/ x (fma (exp (* (* 0.8333333333333334 (- b c)) -2.0)) y x))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))))))) <= 0.0) {
		tmp = 1.0;
	} else {
		tmp = x / fma(exp(((0.8333333333333334 * (b - c)) * -2.0)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) - Float64(Float64(b - c) * Float64(Float64(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0))))))) <= 0.0)
		tmp = 1.0;
	else
		tmp = Float64(x / fma(exp(Float64(Float64(0.8333333333333334 * Float64(b - c)) * -2.0)), y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], 1.0, N[(x / N[(N[Exp[N[(N[(0.8333333333333334 * N[(b - c), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))))) < 0.0

   1. Initial program 99.0%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{1} \]

   if 0.0 < (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))

   1. Initial program 92.2%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} + x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} \cdot y} + x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)}} \]

      4. exp-prod N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\color{blue}{\left(e^{-2}\right)}}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)} \cdot \left(b - c\right)\right)}, y, x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3}}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      14. lower--.f64 87.0

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \color{blue}{\left(b - c\right)}\right)}, y, x\right)} \]

   5. Applied rewrites 87.0%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.8%

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 67.8%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]

   11. Taylor expanded in a around 0

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(\frac{5}{6} \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 58.5%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(0.8333333333333334 \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 74.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)} \leq 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{2 \cdot \left(c \cdot \left(0.8333333333333334 + a\right)\right)}, y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (exp
       (*
        2.0
        (-
         (/ (* z (sqrt (+ t a))) t)
         (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))
      0.0)
   1.0
   (/ x (fma (exp (* 2.0 (* c (+ 0.8333333333333334 a)))) y x))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))))))) <= 0.0) {
		tmp = 1.0;
	} else {
		tmp = x / fma(exp((2.0 * (c * (0.8333333333333334 + a)))), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) - Float64(Float64(b - c) * Float64(Float64(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0))))))) <= 0.0)
		tmp = 1.0;
	else
		tmp = Float64(x / fma(exp(Float64(2.0 * Float64(c * Float64(0.8333333333333334 + a)))), y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], 1.0, N[(x / N[(N[Exp[N[(2.0 * N[(c * N[(0.8333333333333334 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))))) < 0.0

   1. Initial program 99.0%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{1} \]

   if 0.0 < (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))

   1. Initial program 92.2%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} + x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} \cdot y} + x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)}} \]

      4. exp-prod N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\color{blue}{\left(e^{-2}\right)}}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)} \cdot \left(b - c\right)\right)}, y, x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3}}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      14. lower--.f64 87.0

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \color{blue}{\left(b - c\right)}\right)}, y, x\right)} \]

   5. Applied rewrites 87.0%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\mathsf{fma}\left(e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 64.0%

      \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(2 \cdot c\right) \cdot \left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)}, y, x\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \frac{x}{\mathsf{fma}\left(e^{2 \cdot \left(c \cdot \left(\frac{5}{6} + a\right)\right)}, y, x\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 54.2%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{2 \cdot \left(c \cdot \left(0.8333333333333334 + a\right)\right)}, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 71.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)} \leq 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{2 \cdot \left(a \cdot c\right)}, y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (exp
       (*
        2.0
        (-
         (/ (* z (sqrt (+ t a))) t)
         (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))
      0.0)
   1.0
   (/ x (fma (exp (* 2.0 (* a c))) y x))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))))))) <= 0.0) {
		tmp = 1.0;
	} else {
		tmp = x / fma(exp((2.0 * (a * c))), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) - Float64(Float64(b - c) * Float64(Float64(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0))))))) <= 0.0)
		tmp = 1.0;
	else
		tmp = Float64(x / fma(exp(Float64(2.0 * Float64(a * c))), y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], 1.0, N[(x / N[(N[Exp[N[(2.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))))) < 0.0

   1. Initial program 99.0%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{1} \]

   if 0.0 < (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))

   1. Initial program 92.2%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} + x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} \cdot y} + x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)}} \]

      4. exp-prod N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\color{blue}{\left(e^{-2}\right)}}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)} \cdot \left(b - c\right)\right)}, y, x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3}}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      14. lower--.f64 87.0

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \color{blue}{\left(b - c\right)}\right)}, y, x\right)} \]

   5. Applied rewrites 87.0%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\mathsf{fma}\left(e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 64.0%

      \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(2 \cdot c\right) \cdot \left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)}, y, x\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \frac{x}{\mathsf{fma}\left(e^{2 \cdot \left(a \cdot c\right)}, y, x\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 49.2%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{2 \cdot \left(a \cdot c\right)}, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{\left(0.8333333333333334 \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{\left(a \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (-
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))
   (if (<= t_1 -5e+22)
     1.0
     (if (<= t_1 2e+283)
       (/ x (fma (exp (* (* 0.8333333333333334 (- b c)) -2.0)) y x))
       (/ x (fma (exp (* (* a (- b c)) -2.0)) y x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= -5e+22) {
		tmp = 1.0;
	} else if (t_1 <= 2e+283) {
		tmp = x / fma(exp(((0.8333333333333334 * (b - c)) * -2.0)), y, x);
	} else {
		tmp = x / fma(exp(((a * (b - c)) * -2.0)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) - Float64(Float64(b - c) * Float64(Float64(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0)))))
	tmp = 0.0
	if (t_1 <= -5e+22)
		tmp = 1.0;
	elseif (t_1 <= 2e+283)
		tmp = Float64(x / fma(exp(Float64(Float64(0.8333333333333334 * Float64(b - c)) * -2.0)), y, x));
	else
		tmp = Float64(x / fma(exp(Float64(Float64(a * Float64(b - c)) * -2.0)), y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+22], 1.0, If[LessEqual[t$95$1, 2e+283], N[(x / N[(N[Exp[N[(N[(0.8333333333333334 * N[(b - c), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[Exp[N[(N[(a * N[(b - c), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < -4.9999999999999996e22

   1. Initial program 99.0%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{1} \]

   if -4.9999999999999996e22 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < 1.99999999999999991e283

   1. Initial program 99.8%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} + x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} \cdot y} + x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)}} \]

      4. exp-prod N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\color{blue}{\left(e^{-2}\right)}}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)} \cdot \left(b - c\right)\right)}, y, x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3}}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      14. lower--.f64 92.5

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \color{blue}{\left(b - c\right)}\right)}, y, x\right)} \]

   5. Applied rewrites 92.5%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 77.0%

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 77.0%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]

   11. Taylor expanded in a around 0

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(\frac{5}{6} \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 68.1%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(0.8333333333333334 \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]

   if 1.99999999999999991e283 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))

   1. Initial program 85.2%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} + x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} \cdot y} + x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)}} \]

      4. exp-prod N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\color{blue}{\left(e^{-2}\right)}}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)} \cdot \left(b - c\right)\right)}, y, x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3}}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      14. lower--.f64 82.1

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \color{blue}{\left(b - c\right)}\right)}, y, x\right)} \]

   5. Applied rewrites 82.1%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.4%

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 59.4%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]

   11. Taylor expanded in a around inf

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(a \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 63.0%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(a \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-218}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\sqrt{a}, z, 0.6666666666666666 \cdot \left(b - c\right)\right)}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(\sqrt{{t}^{-1}}, z, \frac{b - c}{t} \cdot 0.6666666666666666\right) - \left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= t 5e-218)
   (/
    x
    (+
     x
     (*
      y
      (exp (* 2.0 (/ (fma (sqrt a) z (* 0.6666666666666666 (- b c))) t))))))
   (/
    x
    (+
     x
     (*
      y
      (exp
       (*
        2.0
        (-
         (fma (sqrt (pow t -1.0)) z (* (/ (- b c) t) 0.6666666666666666))
         (* (+ 0.8333333333333334 a) (- b c))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (t <= 5e-218) {
		tmp = x / (x + (y * exp((2.0 * (fma(sqrt(a), z, (0.6666666666666666 * (b - c))) / t)))));
	} else {
		tmp = x / (x + (y * exp((2.0 * (fma(sqrt(pow(t, -1.0)), z, (((b - c) / t) * 0.6666666666666666)) - ((0.8333333333333334 + a) * (b - c)))))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (t <= 5e-218)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(fma(sqrt(a), z, Float64(0.6666666666666666 * Float64(b - c))) / t))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(fma(sqrt((t ^ -1.0)), z, Float64(Float64(Float64(b - c) / t) * 0.6666666666666666)) - Float64(Float64(0.8333333333333334 + a) * Float64(b - c))))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[t, 5e-218], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[Sqrt[a], $MachinePrecision] * z + N[(0.6666666666666666 * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[Sqrt[N[Power[t, -1.0], $MachinePrecision]], $MachinePrecision] * z + N[(N[(N[(b - c), $MachinePrecision] / t), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision] - N[(N[(0.8333333333333334 + a), $MachinePrecision] * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.00000000000000041e-218

   1. Initial program 88.4%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{\sqrt{a} \cdot z - \frac{-2}{3} \cdot \left(b - c\right)}{t}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{\sqrt{a} \cdot z - \frac{-2}{3} \cdot \left(b - c\right)}{t}}}} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \frac{\color{blue}{\sqrt{a} \cdot z + \left(\mathsf{neg}\left(\frac{-2}{3}\right)\right) \cdot \left(b - c\right)}}{t}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \frac{\sqrt{a} \cdot z + \color{blue}{\frac{2}{3}} \cdot \left(b - c\right)}{t}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \frac{\color{blue}{\mathsf{fma}\left(\sqrt{a}, z, \frac{2}{3} \cdot \left(b - c\right)\right)}}{t}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\color{blue}{\sqrt{a}}, z, \frac{2}{3} \cdot \left(b - c\right)\right)}{t}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\sqrt{a}, z, \color{blue}{\frac{2}{3} \cdot \left(b - c\right)}\right)}{t}}} \]

      7. lower--.f64 91.6

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\sqrt{a}, z, 0.6666666666666666 \cdot \color{blue}{\left(b - c\right)}\right)}{t}}} \]

   5. Applied rewrites 91.6%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{\mathsf{fma}\left(\sqrt{a}, z, 0.6666666666666666 \cdot \left(b - c\right)\right)}{t}}}} \]

   if 5.00000000000000041e-218 < t

   1. Initial program 97.3%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\frac{2}{3} \cdot \frac{b - c}{t} + \sqrt{\frac{1}{t}} \cdot z\right) - \left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}}} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\frac{2}{3} \cdot \frac{b - c}{t} + \sqrt{\frac{1}{t}} \cdot z\right) - \left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\left(\sqrt{\frac{1}{t}} \cdot z + \frac{2}{3} \cdot \frac{b - c}{t}\right)} - \left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{t}}, z, \frac{2}{3} \cdot \frac{b - c}{t}\right)} - \left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{t}}}, z, \frac{2}{3} \cdot \frac{b - c}{t}\right) - \left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{t}}}, z, \frac{2}{3} \cdot \frac{b - c}{t}\right) - \left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(\sqrt{\frac{1}{t}}, z, \color{blue}{\frac{b - c}{t} \cdot \frac{2}{3}}\right) - \left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(\sqrt{\frac{1}{t}}, z, \color{blue}{\frac{b - c}{t} \cdot \frac{2}{3}}\right) - \left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(\sqrt{\frac{1}{t}}, z, \color{blue}{\frac{b - c}{t}} \cdot \frac{2}{3}\right) - \left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(\sqrt{\frac{1}{t}}, z, \frac{\color{blue}{b - c}}{t} \cdot \frac{2}{3}\right) - \left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(\sqrt{\frac{1}{t}}, z, \frac{b - c}{t} \cdot \frac{2}{3}\right) - \color{blue}{\left(\frac{5}{6} + a\right) \cdot \left(b - c\right)}\right)}} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(\sqrt{\frac{1}{t}}, z, \frac{b - c}{t} \cdot \frac{2}{3}\right) - \color{blue}{\left(\frac{5}{6} + a\right)} \cdot \left(b - c\right)\right)}} \]

      12. lower--.f64 95.7

          \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(\sqrt{\frac{1}{t}}, z, \frac{b - c}{t} \cdot 0.6666666666666666\right) - \left(0.8333333333333334 + a\right) \cdot \color{blue}{\left(b - c\right)}\right)}} \]

   5. Applied rewrites 95.7%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{\frac{1}{t}}, z, \frac{b - c}{t} \cdot 0.6666666666666666\right) - \left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-218}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\sqrt{a}, z, 0.6666666666666666 \cdot \left(b - c\right)\right)}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(\sqrt{{t}^{-1}}, z, \frac{b - c}{t} \cdot 0.6666666666666666\right) - \left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -2 \cdot \left(b - c\right)\\ \mathbf{if}\;t \leq 5 \cdot 10^{-218}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\sqrt{a}, z, 0.6666666666666666 \cdot \left(b - c\right)\right)}{t}}}\\ \mathbf{elif}\;t \leq 0.8:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{t\_1 \cdot \left(0.8333333333333334 + a\right) - t\_1 \cdot \frac{0.6666666666666666}{t}}, y, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\sqrt{{t}^{-1}}, z, \left(-\left(b - c\right)\right) \cdot \left(0.8333333333333334 + a\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* -2.0 (- b c))))
   (if (<= t 5e-218)
     (/
      x
      (+
       x
       (*
        y
        (exp (* 2.0 (/ (fma (sqrt a) z (* 0.6666666666666666 (- b c))) t))))))
     (if (<= t 0.8)
       (/
        x
        (fma
         (exp
          (-
           (* t_1 (+ 0.8333333333333334 a))
           (* t_1 (/ 0.6666666666666666 t))))
         y
         x))
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (fma
             (sqrt (pow t -1.0))
             z
             (* (- (- b c)) (+ 0.8333333333333334 a))))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = -2.0 * (b - c);
	double tmp;
	if (t <= 5e-218) {
		tmp = x / (x + (y * exp((2.0 * (fma(sqrt(a), z, (0.6666666666666666 * (b - c))) / t)))));
	} else if (t <= 0.8) {
		tmp = x / fma(exp(((t_1 * (0.8333333333333334 + a)) - (t_1 * (0.6666666666666666 / t)))), y, x);
	} else {
		tmp = x / (x + (y * exp((2.0 * fma(sqrt(pow(t, -1.0)), z, (-(b - c) * (0.8333333333333334 + a)))))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(-2.0 * Float64(b - c))
	tmp = 0.0
	if (t <= 5e-218)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(fma(sqrt(a), z, Float64(0.6666666666666666 * Float64(b - c))) / t))))));
	elseif (t <= 0.8)
		tmp = Float64(x / fma(exp(Float64(Float64(t_1 * Float64(0.8333333333333334 + a)) - Float64(t_1 * Float64(0.6666666666666666 / t)))), y, x));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * fma(sqrt((t ^ -1.0)), z, Float64(Float64(-Float64(b - c)) * Float64(0.8333333333333334 + a))))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(-2.0 * N[(b - c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 5e-218], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[Sqrt[a], $MachinePrecision] * z + N[(0.6666666666666666 * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.8], N[(x / N[(N[Exp[N[(N[(t$95$1 * N[(0.8333333333333334 + a), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[Sqrt[N[Power[t, -1.0], $MachinePrecision]], $MachinePrecision] * z + N[((-N[(b - c), $MachinePrecision]) * N[(0.8333333333333334 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.00000000000000041e-218

   1. Initial program 88.4%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{\sqrt{a} \cdot z - \frac{-2}{3} \cdot \left(b - c\right)}{t}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{\sqrt{a} \cdot z - \frac{-2}{3} \cdot \left(b - c\right)}{t}}}} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \frac{\color{blue}{\sqrt{a} \cdot z + \left(\mathsf{neg}\left(\frac{-2}{3}\right)\right) \cdot \left(b - c\right)}}{t}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \frac{\sqrt{a} \cdot z + \color{blue}{\frac{2}{3}} \cdot \left(b - c\right)}{t}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \frac{\color{blue}{\mathsf{fma}\left(\sqrt{a}, z, \frac{2}{3} \cdot \left(b - c\right)\right)}}{t}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\color{blue}{\sqrt{a}}, z, \frac{2}{3} \cdot \left(b - c\right)\right)}{t}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\sqrt{a}, z, \color{blue}{\frac{2}{3} \cdot \left(b - c\right)}\right)}{t}}} \]

      7. lower--.f64 91.6

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\sqrt{a}, z, 0.6666666666666666 \cdot \color{blue}{\left(b - c\right)}\right)}{t}}} \]

   5. Applied rewrites 91.6%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{\mathsf{fma}\left(\sqrt{a}, z, 0.6666666666666666 \cdot \left(b - c\right)\right)}{t}}}} \]

   if 5.00000000000000041e-218 < t < 0.80000000000000004

   1. Initial program 95.6%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} + x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} \cdot y} + x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)}} \]

      4. exp-prod N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\color{blue}{\left(e^{-2}\right)}}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)} \cdot \left(b - c\right)\right)}, y, x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3}}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      14. lower--.f64 87.4

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \color{blue}{\left(b - c\right)}\right)}, y, x\right)} \]

   5. Applied rewrites 87.4%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.8%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left({\left({\left(e^{-2}\right)}^{\left(b - c\right)}\right)}^{\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)}, y, x\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 80.1%

      \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(-2 \cdot \left(b - c\right)\right) \cdot \left(0.8333333333333334 + a\right) - \left(-2 \cdot \left(b - c\right)\right) \cdot \frac{0.6666666666666666}{t}}, y, x\right)} \]

   if 0.80000000000000004 < t

   1. Initial program 98.2%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\sqrt{\frac{1}{t}} \cdot z - \left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{\frac{1}{t}} \cdot z - \color{blue}{\left(b - c\right) \cdot \left(\frac{5}{6} + a\right)}\right)}} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\sqrt{\frac{1}{t}} \cdot z + \left(\mathsf{neg}\left(\left(b - c\right)\right)\right) \cdot \left(\frac{5}{6} + a\right)\right)}}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{t}}, z, \left(\mathsf{neg}\left(\left(b - c\right)\right)\right) \cdot \left(\frac{5}{6} + a\right)\right)}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{t}}}, z, \left(\mathsf{neg}\left(\left(b - c\right)\right)\right) \cdot \left(\frac{5}{6} + a\right)\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{t}}}, z, \left(\mathsf{neg}\left(\left(b - c\right)\right)\right) \cdot \left(\frac{5}{6} + a\right)\right)}} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\sqrt{\frac{1}{t}}, z, \color{blue}{\left(-1 \cdot \left(b - c\right)\right)} \cdot \left(\frac{5}{6} + a\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\sqrt{\frac{1}{t}}, z, \color{blue}{\left(-1 \cdot \left(b - c\right)\right) \cdot \left(\frac{5}{6} + a\right)}\right)}} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\sqrt{\frac{1}{t}}, z, \color{blue}{\left(\mathsf{neg}\left(\left(b - c\right)\right)\right)} \cdot \left(\frac{5}{6} + a\right)\right)}} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\sqrt{\frac{1}{t}}, z, \color{blue}{\left(-\left(b - c\right)\right)} \cdot \left(\frac{5}{6} + a\right)\right)}} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\sqrt{\frac{1}{t}}, z, \left(-\color{blue}{\left(b - c\right)}\right) \cdot \left(\frac{5}{6} + a\right)\right)}} \]

      11. lower-+.f64 99.9

          \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\sqrt{\frac{1}{t}}, z, \left(-\left(b - c\right)\right) \cdot \color{blue}{\left(0.8333333333333334 + a\right)}\right)}} \]

   5. Applied rewrites 99.9%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{t}}, z, \left(-\left(b - c\right)\right) \cdot \left(0.8333333333333334 + a\right)\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-218}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\sqrt{a}, z, 0.6666666666666666 \cdot \left(b - c\right)\right)}{t}}}\\ \mathbf{elif}\;t \leq 0.8:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{\left(-2 \cdot \left(b - c\right)\right) \cdot \left(0.8333333333333334 + a\right) - \left(-2 \cdot \left(b - c\right)\right) \cdot \frac{0.6666666666666666}{t}}, y, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\sqrt{{t}^{-1}}, z, \left(-\left(b - c\right)\right) \cdot \left(0.8333333333333334 + a\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -2 \cdot \left(b - c\right)\\ \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \leq -2 \cdot 10^{+67}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{t\_1 \cdot \left(0.8333333333333334 + a\right) - t\_1 \cdot \frac{0.6666666666666666}{t}}, y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* -2.0 (- b c))))
   (if (<=
        (-
         (/ (* z (sqrt (+ t a))) t)
         (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))
        -2e+67)
     1.0
     (/
      x
      (fma
       (exp
        (- (* t_1 (+ 0.8333333333333334 a)) (* t_1 (/ 0.6666666666666666 t))))
       y
       x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = -2.0 * (b - c);
	double tmp;
	if ((((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))))) <= -2e+67) {
		tmp = 1.0;
	} else {
		tmp = x / fma(exp(((t_1 * (0.8333333333333334 + a)) - (t_1 * (0.6666666666666666 / t)))), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(-2.0 * Float64(b - c))
	tmp = 0.0
	if (Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) - Float64(Float64(b - c) * Float64(Float64(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0))))) <= -2e+67)
		tmp = 1.0;
	else
		tmp = Float64(x / fma(exp(Float64(Float64(t_1 * Float64(0.8333333333333334 + a)) - Float64(t_1 * Float64(0.6666666666666666 / t)))), y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(-2.0 * N[(b - c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e+67], 1.0, N[(x / N[(N[Exp[N[(N[(t$95$1 * N[(0.8333333333333334 + a), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < -1.99999999999999997e67

   1. Initial program 99.0%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{1} \]

   if -1.99999999999999997e67 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))

   1. Initial program 92.5%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} + x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} \cdot y} + x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)}} \]

      4. exp-prod N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\color{blue}{\left(e^{-2}\right)}}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)} \cdot \left(b - c\right)\right)}, y, x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3}}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      14. lower--.f64 87.5

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \color{blue}{\left(b - c\right)}\right)}, y, x\right)} \]

   5. Applied rewrites 87.5%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left({\left({\left(e^{-2}\right)}^{\left(b - c\right)}\right)}^{\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)}, y, x\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 84.4%

      \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(-2 \cdot \left(b - c\right)\right) \cdot \left(0.8333333333333334 + a\right) - \left(-2 \cdot \left(b - c\right)\right) \cdot \frac{0.6666666666666666}{t}}, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 82.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \leq -5 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (-
       (/ (* z (sqrt (+ t a))) t)
       (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))
      -5e+22)
   1.0
   (/ x (fma (exp (* (* (+ 0.8333333333333334 a) (- b c)) -2.0)) y x))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))))) <= -5e+22) {
		tmp = 1.0;
	} else {
		tmp = x / fma(exp((((0.8333333333333334 + a) * (b - c)) * -2.0)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) - Float64(Float64(b - c) * Float64(Float64(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0))))) <= -5e+22)
		tmp = 1.0;
	else
		tmp = Float64(x / fma(exp(Float64(Float64(Float64(0.8333333333333334 + a) * Float64(b - c)) * -2.0)), y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+22], 1.0, N[(x / N[(N[Exp[N[(N[(N[(0.8333333333333334 + a), $MachinePrecision] * N[(b - c), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < -4.9999999999999996e22

   1. Initial program 99.0%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{1} \]

   if -4.9999999999999996e22 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))

   1. Initial program 92.2%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} + x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} \cdot y} + x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)}} \]

      4. exp-prod N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\color{blue}{\left(e^{-2}\right)}}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)} \cdot \left(b - c\right)\right)}, y, x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3}}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      14. lower--.f64 87.0

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \color{blue}{\left(b - c\right)}\right)}, y, x\right)} \]

   5. Applied rewrites 87.0%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.8%

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 67.8%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 85.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)}\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{+140}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{\mathsf{fma}\left(-1.6666666666666667, b - c, 1.3333333333333333 \cdot \frac{b - c}{t}\right)}, y, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{\left(a \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= a -4e-62)
   (/ x (fma (exp (* (* (+ 0.8333333333333334 a) (- b c)) -2.0)) y x))
   (if (<= a 3.6e+140)
     (/
      x
      (fma
       (exp
        (fma -1.6666666666666667 (- b c) (* 1.3333333333333333 (/ (- b c) t))))
       y
       x))
     (/ x (fma (exp (* (* a (- b c)) -2.0)) y x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (a <= -4e-62) {
		tmp = x / fma(exp((((0.8333333333333334 + a) * (b - c)) * -2.0)), y, x);
	} else if (a <= 3.6e+140) {
		tmp = x / fma(exp(fma(-1.6666666666666667, (b - c), (1.3333333333333333 * ((b - c) / t)))), y, x);
	} else {
		tmp = x / fma(exp(((a * (b - c)) * -2.0)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (a <= -4e-62)
		tmp = Float64(x / fma(exp(Float64(Float64(Float64(0.8333333333333334 + a) * Float64(b - c)) * -2.0)), y, x));
	elseif (a <= 3.6e+140)
		tmp = Float64(x / fma(exp(fma(-1.6666666666666667, Float64(b - c), Float64(1.3333333333333333 * Float64(Float64(b - c) / t)))), y, x));
	else
		tmp = Float64(x / fma(exp(Float64(Float64(a * Float64(b - c)) * -2.0)), y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[a, -4e-62], N[(x / N[(N[Exp[N[(N[(N[(0.8333333333333334 + a), $MachinePrecision] * N[(b - c), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.6e+140], N[(x / N[(N[Exp[N[(-1.6666666666666667 * N[(b - c), $MachinePrecision] + N[(1.3333333333333333 * N[(N[(b - c), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[Exp[N[(N[(a * N[(b - c), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.0000000000000002e-62

   1. Initial program 91.1%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} + x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} \cdot y} + x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)}} \]

      4. exp-prod N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\color{blue}{\left(e^{-2}\right)}}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)} \cdot \left(b - c\right)\right)}, y, x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3}}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      14. lower--.f64 86.8

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \color{blue}{\left(b - c\right)}\right)}, y, x\right)} \]

   5. Applied rewrites 86.8%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 91.2%

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 91.2%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]

   if -4.0000000000000002e-62 < a < 3.6e140

   1. Initial program 97.0%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} + x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} \cdot y} + x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)}} \]

      4. exp-prod N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\color{blue}{\left(e^{-2}\right)}}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)} \cdot \left(b - c\right)\right)}, y, x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3}}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      14. lower--.f64 89.4

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \color{blue}{\left(b - c\right)}\right)}, y, x\right)} \]

   5. Applied rewrites 89.4%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left({\left({\left(e^{-2}\right)}^{\left(b - c\right)}\right)}^{\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)}, y, x\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 87.1%

      \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(-2 \cdot \left(b - c\right)\right) \cdot \left(0.8333333333333334 + a\right) - \left(-2 \cdot \left(b - c\right)\right) \cdot \frac{0.6666666666666666}{t}}, y, x\right)} \]

   10. Taylor expanded in a around 0

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\frac{-5}{3} \cdot \left(b - c\right) - \frac{-4}{3} \cdot \frac{b - c}{t}}, y, x\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 90.0%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\mathsf{fma}\left(-1.6666666666666667, b - c, 1.3333333333333333 \cdot \frac{b - c}{t}\right)}, y, x\right)} \]

   if 3.6e140 < a

   1. Initial program 90.3%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} + x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} \cdot y} + x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)}} \]

      4. exp-prod N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\color{blue}{\left(e^{-2}\right)}}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)} \cdot \left(b - c\right)\right)}, y, x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3}}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      14. lower--.f64 84.4

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \color{blue}{\left(b - c\right)}\right)}, y, x\right)} \]

   5. Applied rewrites 84.4%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 84.4%

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 84.4%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]

   11. Taylor expanded in a around inf

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(a \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 84.4%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(a \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 84.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{\left(a \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{1.3333333333333333 \cdot \frac{b - c}{t}}, y, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= t -2.5e-106)
   (/ x (fma (exp (* (* a (- b c)) -2.0)) y x))
   (if (<= t 2.5e-5)
     (/ x (fma (exp (* 1.3333333333333333 (/ (- b c) t))) y x))
     (/ x (fma (exp (* (* (+ 0.8333333333333334 a) (- b c)) -2.0)) y x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (t <= -2.5e-106) {
		tmp = x / fma(exp(((a * (b - c)) * -2.0)), y, x);
	} else if (t <= 2.5e-5) {
		tmp = x / fma(exp((1.3333333333333333 * ((b - c) / t))), y, x);
	} else {
		tmp = x / fma(exp((((0.8333333333333334 + a) * (b - c)) * -2.0)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (t <= -2.5e-106)
		tmp = Float64(x / fma(exp(Float64(Float64(a * Float64(b - c)) * -2.0)), y, x));
	elseif (t <= 2.5e-5)
		tmp = Float64(x / fma(exp(Float64(1.3333333333333333 * Float64(Float64(b - c) / t))), y, x));
	else
		tmp = Float64(x / fma(exp(Float64(Float64(Float64(0.8333333333333334 + a) * Float64(b - c)) * -2.0)), y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[t, -2.5e-106], N[(x / N[(N[Exp[N[(N[(a * N[(b - c), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.5e-5], N[(x / N[(N[Exp[N[(1.3333333333333333 * N[(N[(b - c), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[Exp[N[(N[(N[(0.8333333333333334 + a), $MachinePrecision] * N[(b - c), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.49999999999999991e-106

   1. Initial program 100.0%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} + x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} \cdot y} + x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)}} \]

      4. exp-prod N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\color{blue}{\left(e^{-2}\right)}}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)} \cdot \left(b - c\right)\right)}, y, x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3}}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      14. lower--.f64 88.8

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \color{blue}{\left(b - c\right)}\right)}, y, x\right)} \]

   5. Applied rewrites 88.8%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 88.8%

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 88.8%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]

   11. Taylor expanded in a around inf

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(a \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 92.5%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(a \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]

   if -2.49999999999999991e-106 < t < 2.50000000000000012e-5

   1. Initial program 90.2%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} + x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} \cdot y} + x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)}} \]

      4. exp-prod N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\color{blue}{\left(e^{-2}\right)}}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)} \cdot \left(b - c\right)\right)}, y, x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3}}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      14. lower--.f64 84.4

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \color{blue}{\left(b - c\right)}\right)}, y, x\right)} \]

   5. Applied rewrites 84.4%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.1%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left({\left({\left(e^{-2}\right)}^{\left(b - c\right)}\right)}^{\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)}, y, x\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 76.4%

      \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(-2 \cdot \left(b - c\right)\right) \cdot \left(0.8333333333333334 + a\right) - \left(-2 \cdot \left(b - c\right)\right) \cdot \frac{0.6666666666666666}{t}}, y, x\right)} \]

   10. Taylor expanded in t around 0

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\frac{4}{3} \cdot \frac{b - c}{t}}, y, x\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 74.0%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{1.3333333333333333 \cdot \frac{b - c}{t}}, y, x\right)} \]

   if 2.50000000000000012e-5 < t

   1. Initial program 98.2%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x}{\color{blue}{x + y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} + x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)} \cdot y} + x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(e^{-2 \cdot \left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)}} \]

      4. exp-prod N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(e^{-2}\right)}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}, y, x\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\color{blue}{\left(e^{-2}\right)}}^{\left(\left(b - c\right) \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}, y, x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\color{blue}{\left(\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}}, y, x\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)} \cdot \left(b - c\right)\right)}, y, x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3}}{t}}\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]

      14. lower--.f64 91.2

          \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \color{blue}{\left(b - c\right)}\right)}, y, x\right)} \]

   5. Applied rewrites 91.2%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}, y, x\right)}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(\frac{5}{6} + a\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 91.2%

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(e^{-2}\right)}^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}, y, x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 91.2%

       \[\leadsto \frac{x}{\mathsf{fma}\left(e^{\left(\left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right) \cdot -2}, y, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 51.6% accurate, 198.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z t a b c) :precision binary64 1.0)

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return 1.0;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return 1.0;
}

Python

def code(x, y, z, t, a, b, c):
	return 1.0

Julia

function code(x, y, z, t, a, b, c)
	return 1.0
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := 1.0

Derivation

1. Initial program 94.9%

   \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 44.8%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Developer Target 1: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \sqrt{t + a}\\ t_2 := a - \frac{5}{6}\\ \mathbf{if}\;t < -2.118326644891581 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t < 5.196588770651547 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{t\_1 \cdot \left(\left(3 \cdot t\right) \cdot t\_2\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(t\_2 \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot t\_2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{t\_1}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* z (sqrt (+ t a)))) (t_2 (- a (/ 5.0 6.0))))
   (if (< t -2.118326644891581e-50)
     (/
      x
      (+
       x
       (* y (exp (* 2.0 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b)))))))
     (if (< t 5.196588770651547e-123)
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (/
             (-
              (* t_1 (* (* 3.0 t) t_2))
              (*
               (- (* (+ (/ 5.0 6.0) a) (* 3.0 t)) 2.0)
               (* t_2 (* (- b c) t))))
             (* (* (* t t) 3.0) t_2)))))))
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (-
             (/ t_1 t)
             (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = z * sqrt((t + a));
	double t_2 = a - (5.0 / 6.0);
	double tmp;
	if (t < -2.118326644891581e-50) {
		tmp = x / (x + (y * exp((2.0 * (((a * c) + (0.8333333333333334 * c)) - (a * b))))));
	} else if (t < 5.196588770651547e-123) {
		tmp = x / (x + (y * exp((2.0 * (((t_1 * ((3.0 * t) * t_2)) - (((((5.0 / 6.0) + a) * (3.0 * t)) - 2.0) * (t_2 * ((b - c) * t)))) / (((t * t) * 3.0) * t_2))))));
	} else {
		tmp = x / (x + (y * exp((2.0 * ((t_1 / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = z * sqrt((t + a))
    t_2 = a - (5.0d0 / 6.0d0)
    if (t < (-2.118326644891581d-50)) then
        tmp = x / (x + (y * exp((2.0d0 * (((a * c) + (0.8333333333333334d0 * c)) - (a * b))))))
    else if (t < 5.196588770651547d-123) then
        tmp = x / (x + (y * exp((2.0d0 * (((t_1 * ((3.0d0 * t) * t_2)) - (((((5.0d0 / 6.0d0) + a) * (3.0d0 * t)) - 2.0d0) * (t_2 * ((b - c) * t)))) / (((t * t) * 3.0d0) * t_2))))))
    else
        tmp = x / (x + (y * exp((2.0d0 * ((t_1 / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0)))))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = z * Math.sqrt((t + a));
	double t_2 = a - (5.0 / 6.0);
	double tmp;
	if (t < -2.118326644891581e-50) {
		tmp = x / (x + (y * Math.exp((2.0 * (((a * c) + (0.8333333333333334 * c)) - (a * b))))));
	} else if (t < 5.196588770651547e-123) {
		tmp = x / (x + (y * Math.exp((2.0 * (((t_1 * ((3.0 * t) * t_2)) - (((((5.0 / 6.0) + a) * (3.0 * t)) - 2.0) * (t_2 * ((b - c) * t)))) / (((t * t) * 3.0) * t_2))))));
	} else {
		tmp = x / (x + (y * Math.exp((2.0 * ((t_1 / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = z * math.sqrt((t + a))
	t_2 = a - (5.0 / 6.0)
	tmp = 0
	if t < -2.118326644891581e-50:
		tmp = x / (x + (y * math.exp((2.0 * (((a * c) + (0.8333333333333334 * c)) - (a * b))))))
	elif t < 5.196588770651547e-123:
		tmp = x / (x + (y * math.exp((2.0 * (((t_1 * ((3.0 * t) * t_2)) - (((((5.0 / 6.0) + a) * (3.0 * t)) - 2.0) * (t_2 * ((b - c) * t)))) / (((t * t) * 3.0) * t_2))))))
	else:
		tmp = x / (x + (y * math.exp((2.0 * ((t_1 / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(z * sqrt(Float64(t + a)))
	t_2 = Float64(a - Float64(5.0 / 6.0))
	tmp = 0.0
	if (t < -2.118326644891581e-50)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(a * c) + Float64(0.8333333333333334 * c)) - Float64(a * b)))))));
	elseif (t < 5.196588770651547e-123)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(t_1 * Float64(Float64(3.0 * t) * t_2)) - Float64(Float64(Float64(Float64(Float64(5.0 / 6.0) + a) * Float64(3.0 * t)) - 2.0) * Float64(t_2 * Float64(Float64(b - c) * t)))) / Float64(Float64(Float64(t * t) * 3.0) * t_2)))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(t_1 / t) - Float64(Float64(b - c) * Float64(Float64(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = z * sqrt((t + a));
	t_2 = a - (5.0 / 6.0);
	tmp = 0.0;
	if (t < -2.118326644891581e-50)
		tmp = x / (x + (y * exp((2.0 * (((a * c) + (0.8333333333333334 * c)) - (a * b))))));
	elseif (t < 5.196588770651547e-123)
		tmp = x / (x + (y * exp((2.0 * (((t_1 * ((3.0 * t) * t_2)) - (((((5.0 / 6.0) + a) * (3.0 * t)) - 2.0) * (t_2 * ((b - c) * t)))) / (((t * t) * 3.0) * t_2))))));
	else
		tmp = x / (x + (y * exp((2.0 * ((t_1 / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a - N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -2.118326644891581e-50], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(a * c), $MachinePrecision] + N[(0.8333333333333334 * c), $MachinePrecision]), $MachinePrecision] - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[t, 5.196588770651547e-123], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(t$95$1 * N[(N[(3.0 * t), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision] * N[(3.0 * t), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * N[(t$95$2 * N[(N[(b - c), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t * t), $MachinePrecision] * 3.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(t$95$1 / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024320 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -2118326644891581/100000000000000000000000000000000000000000000000000000000000000000) (/ x (+ x (* y (exp (* 2 (- (+ (* a c) (* 4166666666666667/5000000000000000 c)) (* a b))))))) (if (< t 5196588770651547/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ x (+ x (* y (exp (* 2 (/ (- (* (* z (sqrt (+ t a))) (* (* 3 t) (- a (/ 5 6)))) (* (- (* (+ (/ 5 6) a) (* 3 t)) 2) (* (- a (/ 5 6)) (* (- b c) t)))) (* (* (* t t) 3) (- a (/ 5 6))))))))) (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))))

  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))