Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B

Percentage Accurate: 96.3% → 96.3%

Time: 16.4s

Alternatives: 19

Speedup: 1.0×

• 41.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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
    code = x * exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
}

Python

def code(x, y, z, t, a, b):
	return x * math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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
    code = x * exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
}

Python

def code(x, y, z, t, a, b):
	return x * math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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
    code = x * exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
}

Python

def code(x, y, z, t, a, b):
	return x * math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 54.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+45}:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 50000000000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -0.16666666666666666, x \cdot \left(b \cdot \left(b \cdot b\right)\right), 0.5 \cdot \left(x \cdot \left(b \cdot b\right)\right)\right), x \cdot \left(-b\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, 0.5 \cdot \left(t \cdot \left(y \cdot y\right)\right) - y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -1e+45)
     (* x (* y (* y (* y (* -0.16666666666666666 (* t (* t t)))))))
     (if (<= t_1 50000000000000.0)
       (* x (fma b (fma 0.5 (* b (* a a)) (- a)) 1.0))
       (if (<= t_1 5e+151)
         (fma
          a
          (fma
           a
           (fma
            (* a -0.16666666666666666)
            (* x (* b (* b b)))
            (* 0.5 (* x (* b b))))
           (* x (- b)))
          x)
         (* x (fma t (- (* 0.5 (* t (* y y))) y) 1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -1e+45) {
		tmp = x * (y * (y * (y * (-0.16666666666666666 * (t * (t * t))))));
	} else if (t_1 <= 50000000000000.0) {
		tmp = x * fma(b, fma(0.5, (b * (a * a)), -a), 1.0);
	} else if (t_1 <= 5e+151) {
		tmp = fma(a, fma(a, fma((a * -0.16666666666666666), (x * (b * (b * b))), (0.5 * (x * (b * b)))), (x * -b)), x);
	} else {
		tmp = x * fma(t, ((0.5 * (t * (y * y))) - y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -1e+45)
		tmp = Float64(x * Float64(y * Float64(y * Float64(y * Float64(-0.16666666666666666 * Float64(t * Float64(t * t)))))));
	elseif (t_1 <= 50000000000000.0)
		tmp = Float64(x * fma(b, fma(0.5, Float64(b * Float64(a * a)), Float64(-a)), 1.0));
	elseif (t_1 <= 5e+151)
		tmp = fma(a, fma(a, fma(Float64(a * -0.16666666666666666), Float64(x * Float64(b * Float64(b * b))), Float64(0.5 * Float64(x * Float64(b * b)))), Float64(x * Float64(-b))), x);
	else
		tmp = Float64(x * fma(t, Float64(Float64(0.5 * Float64(t * Float64(y * y))) - y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+45], N[(x * N[(y * N[(y * N[(y * N[(-0.16666666666666666 * N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 50000000000000.0], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+151], N[(a * N[(a * N[(N[(a * -0.16666666666666666), $MachinePrecision] * N[(x * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * (-b)), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x * N[(t * N[(N[(0.5 * N[(t * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -9.9999999999999993e44

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 53.7

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 53.7%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. cube-mult N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{-1}{6}, t \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)}, \frac{1}{2} \cdot {y}^{2}\right), -1 \cdot y\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{-1}{6}, t \cdot \left(y \cdot \color{blue}{{y}^{2}}\right), \frac{1}{2} \cdot {y}^{2}\right), -1 \cdot y\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{-1}{6}, t \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}, \frac{1}{2} \cdot {y}^{2}\right), -1 \cdot y\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{-1}{6}, t \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right), \frac{1}{2} \cdot {y}^{2}\right), -1 \cdot y\right), 1\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{-1}{6}, t \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right), \frac{1}{2} \cdot {y}^{2}\right), -1 \cdot y\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{-1}{6}, t \cdot \left(y \cdot \left(y \cdot y\right)\right), \color{blue}{\frac{1}{2} \cdot {y}^{2}}\right), -1 \cdot y\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{-1}{6}, t \cdot \left(y \cdot \left(y \cdot y\right)\right), \frac{1}{2} \cdot \color{blue}{\left(y \cdot y\right)}\right), -1 \cdot y\right), 1\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{-1}{6}, t \cdot \left(y \cdot \left(y \cdot y\right)\right), \frac{1}{2} \cdot \color{blue}{\left(y \cdot y\right)}\right), -1 \cdot y\right), 1\right) \]

      15. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{-1}{6}, t \cdot \left(y \cdot \left(y \cdot y\right)\right), \frac{1}{2} \cdot \left(y \cdot y\right)\right), \color{blue}{\mathsf{neg}\left(y\right)}\right), 1\right) \]

      16. neg-lowering-neg.f64 1.8

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(-0.16666666666666666, t \cdot \left(y \cdot \left(y \cdot y\right)\right), 0.5 \cdot \left(y \cdot y\right)\right), \color{blue}{-y}\right), 1\right) \]

   8. Simplified 1.8%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(-0.16666666666666666, t \cdot \left(y \cdot \left(y \cdot y\right)\right), 0.5 \cdot \left(y \cdot y\right)\right), -y\right), 1\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({t}^{3} \cdot {y}^{3}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {t}^{3}\right) \cdot {y}^{3}\right)} \]

       2. unpow3 N/A

          \[\leadsto x \cdot \left(\left(\frac{-1}{6} \cdot {t}^{3}\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot y\right)}\right) \]

       3. unpow2 N/A

          \[\leadsto x \cdot \left(\left(\frac{-1}{6} \cdot {t}^{3}\right) \cdot \left(\color{blue}{{y}^{2}} \cdot y\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\left(\frac{-1}{6} \cdot {t}^{3}\right) \cdot {y}^{2}\right) \cdot y\right)} \]

       5. unpow2 N/A

          \[\leadsto x \cdot \left(\left(\left(\frac{-1}{6} \cdot {t}^{3}\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot y\right) \]

       6. associate-*l* N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(\left(\left(\frac{-1}{6} \cdot {t}^{3}\right) \cdot y\right) \cdot y\right)} \cdot y\right) \]

       7. associate-*r* N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{\left(\frac{-1}{6} \cdot \left({t}^{3} \cdot y\right)\right)} \cdot y\right) \cdot y\right) \]

       8. *-commutative N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{\left(\left({t}^{3} \cdot y\right) \cdot \frac{-1}{6}\right)} \cdot y\right) \cdot y\right) \]

       9. *-commutative N/A

          \[\leadsto x \cdot \left(\left(\left(\color{blue}{\left(y \cdot {t}^{3}\right)} \cdot \frac{-1}{6}\right) \cdot y\right) \cdot y\right) \]

       10. associate-*l* N/A

           \[\leadsto x \cdot \left(\left(\color{blue}{\left(y \cdot \left({t}^{3} \cdot \frac{-1}{6}\right)\right)} \cdot y\right) \cdot y\right) \]

       11. *-commutative N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {t}^{3}\right)}\right) \cdot y\right) \cdot y\right) \]

       12. associate-*r* N/A

           \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot \left(\left(\frac{-1}{6} \cdot {t}^{3}\right) \cdot y\right)\right)} \cdot y\right) \]

       13. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(y \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({t}^{3} \cdot y\right)\right)}\right) \cdot y\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left(\left(y \cdot \left(\frac{-1}{6} \cdot \left({t}^{3} \cdot y\right)\right)\right) \cdot y\right)} \]

   11. Simplified 33.8%

       \[\leadsto x \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right) \cdot y\right)} \]

   if -9.9999999999999993e44 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5e13

   1. Initial program 93.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 78.8

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 78.8%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + -1 \cdot a}, 1\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot b, -1 \cdot a\right)}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{b \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{b \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      9. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \left(a \cdot a\right), \color{blue}{\mathsf{neg}\left(a\right)}\right), 1\right) \]

      10. neg-lowering-neg.f64 81.7

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), \color{blue}{-a}\right), 1\right) \]

   8. Simplified 81.7%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)} \]

   if 5e13 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5.0000000000000002e151

   1. Initial program 82.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 45.0

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 45.0%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 36.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-0.16666666666666666 \cdot a, x \cdot \left(b \cdot \left(b \cdot b\right)\right), 0.5 \cdot \left(x \cdot \left(b \cdot b\right)\right)\right), -x \cdot b\right), x\right)} \]

   if 5.0000000000000002e151 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 95.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 61.1

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 61.1%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + t \cdot \left(-1 \cdot y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(t \cdot \left(-1 \cdot y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) + -1 \cdot y}, 1\right) \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, 1\right) \]

      5. unsub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) - y}, 1\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) - y}, 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)} - y, 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \frac{1}{2} \cdot \color{blue}{\left(t \cdot {y}^{2}\right)} - y, 1\right) \]

      9. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \frac{1}{2} \cdot \left(t \cdot \color{blue}{\left(y \cdot y\right)}\right) - y, 1\right) \]

      10. *-lowering-*.f64 74.6

          \[\leadsto x \cdot \mathsf{fma}\left(t, 0.5 \cdot \left(t \cdot \color{blue}{\left(y \cdot y\right)}\right) - y, 1\right) \]

   8. Simplified 74.6%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, 0.5 \cdot \left(t \cdot \left(y \cdot y\right)\right) - y, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -1 \cdot 10^{+45}:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 50000000000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -0.16666666666666666, x \cdot \left(b \cdot \left(b \cdot b\right)\right), 0.5 \cdot \left(x \cdot \left(b \cdot b\right)\right)\right), x \cdot \left(-b\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, 0.5 \cdot \left(t \cdot \left(y \cdot y\right)\right) - y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+45}:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 50000000000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+150}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \left(y \cdot \left(x \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, 0.5 \cdot \left(t \cdot \left(y \cdot y\right)\right) - y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -1e+45)
     (* x (* y (* y (* y (* -0.16666666666666666 (* t (* t t)))))))
     (if (<= t_1 50000000000000.0)
       (* x (fma b (fma 0.5 (* b (* a a)) (- a)) 1.0))
       (if (<= t_1 2e+150)
         (* y (* 0.5 (* y (* x (* t t)))))
         (* x (fma t (- (* 0.5 (* t (* y y))) y) 1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -1e+45) {
		tmp = x * (y * (y * (y * (-0.16666666666666666 * (t * (t * t))))));
	} else if (t_1 <= 50000000000000.0) {
		tmp = x * fma(b, fma(0.5, (b * (a * a)), -a), 1.0);
	} else if (t_1 <= 2e+150) {
		tmp = y * (0.5 * (y * (x * (t * t))));
	} else {
		tmp = x * fma(t, ((0.5 * (t * (y * y))) - y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -1e+45)
		tmp = Float64(x * Float64(y * Float64(y * Float64(y * Float64(-0.16666666666666666 * Float64(t * Float64(t * t)))))));
	elseif (t_1 <= 50000000000000.0)
		tmp = Float64(x * fma(b, fma(0.5, Float64(b * Float64(a * a)), Float64(-a)), 1.0));
	elseif (t_1 <= 2e+150)
		tmp = Float64(y * Float64(0.5 * Float64(y * Float64(x * Float64(t * t)))));
	else
		tmp = Float64(x * fma(t, Float64(Float64(0.5 * Float64(t * Float64(y * y))) - y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+45], N[(x * N[(y * N[(y * N[(y * N[(-0.16666666666666666 * N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 50000000000000.0], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+150], N[(y * N[(0.5 * N[(y * N[(x * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(t * N[(N[(0.5 * N[(t * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -9.9999999999999993e44

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 53.7

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 53.7%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. cube-mult N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{-1}{6}, t \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)}, \frac{1}{2} \cdot {y}^{2}\right), -1 \cdot y\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{-1}{6}, t \cdot \left(y \cdot \color{blue}{{y}^{2}}\right), \frac{1}{2} \cdot {y}^{2}\right), -1 \cdot y\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{-1}{6}, t \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}, \frac{1}{2} \cdot {y}^{2}\right), -1 \cdot y\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{-1}{6}, t \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right), \frac{1}{2} \cdot {y}^{2}\right), -1 \cdot y\right), 1\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{-1}{6}, t \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right), \frac{1}{2} \cdot {y}^{2}\right), -1 \cdot y\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{-1}{6}, t \cdot \left(y \cdot \left(y \cdot y\right)\right), \color{blue}{\frac{1}{2} \cdot {y}^{2}}\right), -1 \cdot y\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{-1}{6}, t \cdot \left(y \cdot \left(y \cdot y\right)\right), \frac{1}{2} \cdot \color{blue}{\left(y \cdot y\right)}\right), -1 \cdot y\right), 1\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{-1}{6}, t \cdot \left(y \cdot \left(y \cdot y\right)\right), \frac{1}{2} \cdot \color{blue}{\left(y \cdot y\right)}\right), -1 \cdot y\right), 1\right) \]

      15. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(\frac{-1}{6}, t \cdot \left(y \cdot \left(y \cdot y\right)\right), \frac{1}{2} \cdot \left(y \cdot y\right)\right), \color{blue}{\mathsf{neg}\left(y\right)}\right), 1\right) \]

      16. neg-lowering-neg.f64 1.8

          \[\leadsto x \cdot \mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(-0.16666666666666666, t \cdot \left(y \cdot \left(y \cdot y\right)\right), 0.5 \cdot \left(y \cdot y\right)\right), \color{blue}{-y}\right), 1\right) \]

   8. Simplified 1.8%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, \mathsf{fma}\left(t, \mathsf{fma}\left(-0.16666666666666666, t \cdot \left(y \cdot \left(y \cdot y\right)\right), 0.5 \cdot \left(y \cdot y\right)\right), -y\right), 1\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({t}^{3} \cdot {y}^{3}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {t}^{3}\right) \cdot {y}^{3}\right)} \]

       2. unpow3 N/A

          \[\leadsto x \cdot \left(\left(\frac{-1}{6} \cdot {t}^{3}\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot y\right)}\right) \]

       3. unpow2 N/A

          \[\leadsto x \cdot \left(\left(\frac{-1}{6} \cdot {t}^{3}\right) \cdot \left(\color{blue}{{y}^{2}} \cdot y\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\left(\frac{-1}{6} \cdot {t}^{3}\right) \cdot {y}^{2}\right) \cdot y\right)} \]

       5. unpow2 N/A

          \[\leadsto x \cdot \left(\left(\left(\frac{-1}{6} \cdot {t}^{3}\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot y\right) \]

       6. associate-*l* N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(\left(\left(\frac{-1}{6} \cdot {t}^{3}\right) \cdot y\right) \cdot y\right)} \cdot y\right) \]

       7. associate-*r* N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{\left(\frac{-1}{6} \cdot \left({t}^{3} \cdot y\right)\right)} \cdot y\right) \cdot y\right) \]

       8. *-commutative N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{\left(\left({t}^{3} \cdot y\right) \cdot \frac{-1}{6}\right)} \cdot y\right) \cdot y\right) \]

       9. *-commutative N/A

          \[\leadsto x \cdot \left(\left(\left(\color{blue}{\left(y \cdot {t}^{3}\right)} \cdot \frac{-1}{6}\right) \cdot y\right) \cdot y\right) \]

       10. associate-*l* N/A

           \[\leadsto x \cdot \left(\left(\color{blue}{\left(y \cdot \left({t}^{3} \cdot \frac{-1}{6}\right)\right)} \cdot y\right) \cdot y\right) \]

       11. *-commutative N/A

           \[\leadsto x \cdot \left(\left(\left(y \cdot \color{blue}{\left(\frac{-1}{6} \cdot {t}^{3}\right)}\right) \cdot y\right) \cdot y\right) \]

       12. associate-*r* N/A

           \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot \left(\left(\frac{-1}{6} \cdot {t}^{3}\right) \cdot y\right)\right)} \cdot y\right) \]

       13. associate-*r* N/A

           \[\leadsto x \cdot \left(\left(y \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({t}^{3} \cdot y\right)\right)}\right) \cdot y\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left(\left(y \cdot \left(\frac{-1}{6} \cdot \left({t}^{3} \cdot y\right)\right)\right) \cdot y\right)} \]

   11. Simplified 33.8%

       \[\leadsto x \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right) \cdot y\right)} \]

   if -9.9999999999999993e44 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5e13

   1. Initial program 93.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 78.8

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 78.8%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + -1 \cdot a}, 1\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot b, -1 \cdot a\right)}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{b \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{b \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      9. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \left(a \cdot a\right), \color{blue}{\mathsf{neg}\left(a\right)}\right), 1\right) \]

      10. neg-lowering-neg.f64 81.7

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), \color{blue}{-a}\right), 1\right) \]

   8. Simplified 81.7%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)} \]

   if 5e13 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.99999999999999996e150

   1. Initial program 82.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 21.3

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 21.3%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-lowering-neg.f64 21.5

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{-t}\right), 1\right) \]

   8. Simplified 21.5%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{t \cdot \left(\frac{1}{2} \cdot \left(t \cdot y\right) - 1\right)}, 1\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{t \cdot \left(\frac{1}{2} \cdot \left(t \cdot y\right) - 1\right)}, 1\right) \]

       2. sub-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, t \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(t \cdot y\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}, 1\right) \]

       3. metadata-eval N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, t \cdot \left(\frac{1}{2} \cdot \left(t \cdot y\right) + \color{blue}{-1}\right), 1\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, t \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, t \cdot y, -1\right)}, 1\right) \]

       5. *-lowering-*.f64 17.1

          \[\leadsto x \cdot \mathsf{fma}\left(y, t \cdot \mathsf{fma}\left(0.5, \color{blue}{t \cdot y}, -1\right), 1\right) \]

   11. Simplified 17.1%

       \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{t \cdot \mathsf{fma}\left(0.5, t \cdot y, -1\right)}, 1\right) \]

   12. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left({t}^{2} \cdot x\right) \cdot {y}^{2}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot {y}^{2}} \]

       3. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot \color{blue}{\left(y \cdot y\right)} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot y\right) \cdot y} \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{y \cdot \left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot y\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot y\right)} \]

       7. associate-*r* N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\left({t}^{2} \cdot x\right) \cdot y\right)\right)} \]

       8. associate-*r* N/A

          \[\leadsto y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({t}^{2} \cdot \left(x \cdot y\right)\right)}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot y\right)\right)\right)} \]

       10. associate-*r* N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left({t}^{2} \cdot x\right) \cdot y\right)}\right) \]

       11. *-commutative N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot \left({t}^{2} \cdot x\right)\right)}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot \left({t}^{2} \cdot x\right)\right)}\right) \]

       13. *-commutative N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \left(y \cdot \color{blue}{\left(x \cdot {t}^{2}\right)}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \left(y \cdot \color{blue}{\left(x \cdot {t}^{2}\right)}\right)\right) \]

       15. unpow2 N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \left(y \cdot \left(x \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

       16. *-lowering-*.f64 37.8

           \[\leadsto y \cdot \left(0.5 \cdot \left(y \cdot \left(x \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

   14. Simplified 37.8%

       \[\leadsto \color{blue}{y \cdot \left(0.5 \cdot \left(y \cdot \left(x \cdot \left(t \cdot t\right)\right)\right)\right)} \]

   if 1.99999999999999996e150 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 95.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 60.4

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 60.4%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + t \cdot \left(-1 \cdot y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(t \cdot \left(-1 \cdot y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) + -1 \cdot y}, 1\right) \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, 1\right) \]

      5. unsub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) - y}, 1\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) - y}, 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)} - y, 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \frac{1}{2} \cdot \color{blue}{\left(t \cdot {y}^{2}\right)} - y, 1\right) \]

      9. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \frac{1}{2} \cdot \left(t \cdot \color{blue}{\left(y \cdot y\right)}\right) - y, 1\right) \]

      10. *-lowering-*.f64 73.7

          \[\leadsto x \cdot \mathsf{fma}\left(t, 0.5 \cdot \left(t \cdot \color{blue}{\left(y \cdot y\right)}\right) - y, 1\right) \]

   8. Simplified 73.7%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, 0.5 \cdot \left(t \cdot \left(y \cdot y\right)\right) - y, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -1 \cdot 10^{+45}:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 50000000000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+150}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \left(y \cdot \left(x \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, 0.5 \cdot \left(t \cdot \left(y \cdot y\right)\right) - y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(0.5 \cdot \left(y \cdot \left(x \cdot \left(t \cdot t\right)\right)\right)\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 50000000000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, 0.5 \cdot \left(t \cdot \left(y \cdot y\right)\right) - y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* y (* 0.5 (* y (* x (* t t))))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -1e+45)
     t_1
     (if (<= t_2 50000000000000.0)
       (* x (fma b (fma 0.5 (* b (* a a)) (- a)) 1.0))
       (if (<= t_2 2e+150)
         t_1
         (* x (fma t (- (* 0.5 (* t (* y y))) y) 1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y * (0.5 * (y * (x * (t * t))));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -1e+45) {
		tmp = t_1;
	} else if (t_2 <= 50000000000000.0) {
		tmp = x * fma(b, fma(0.5, (b * (a * a)), -a), 1.0);
	} else if (t_2 <= 2e+150) {
		tmp = t_1;
	} else {
		tmp = x * fma(t, ((0.5 * (t * (y * y))) - y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y * Float64(0.5 * Float64(y * Float64(x * Float64(t * t)))))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -1e+45)
		tmp = t_1;
	elseif (t_2 <= 50000000000000.0)
		tmp = Float64(x * fma(b, fma(0.5, Float64(b * Float64(a * a)), Float64(-a)), 1.0));
	elseif (t_2 <= 2e+150)
		tmp = t_1;
	else
		tmp = Float64(x * fma(t, Float64(Float64(0.5 * Float64(t * Float64(y * y))) - y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(0.5 * N[(y * N[(x * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+45], t$95$1, If[LessEqual[t$95$2, 50000000000000.0], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+150], t$95$1, N[(x * N[(t * N[(N[(0.5 * N[(t * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -9.9999999999999993e44 or 5e13 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.99999999999999996e150

   1. Initial program 96.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 47.4

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 47.4%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-lowering-neg.f64 6.1

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{-t}\right), 1\right) \]

   8. Simplified 6.1%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{t \cdot \left(\frac{1}{2} \cdot \left(t \cdot y\right) - 1\right)}, 1\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{t \cdot \left(\frac{1}{2} \cdot \left(t \cdot y\right) - 1\right)}, 1\right) \]

       2. sub-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, t \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(t \cdot y\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}, 1\right) \]

       3. metadata-eval N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, t \cdot \left(\frac{1}{2} \cdot \left(t \cdot y\right) + \color{blue}{-1}\right), 1\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, t \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, t \cdot y, -1\right)}, 1\right) \]

       5. *-lowering-*.f64 5.3

          \[\leadsto x \cdot \mathsf{fma}\left(y, t \cdot \mathsf{fma}\left(0.5, \color{blue}{t \cdot y}, -1\right), 1\right) \]

   11. Simplified 5.3%

       \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{t \cdot \mathsf{fma}\left(0.5, t \cdot y, -1\right)}, 1\right) \]

   12. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left({t}^{2} \cdot x\right) \cdot {y}^{2}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot {y}^{2}} \]

       3. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot \color{blue}{\left(y \cdot y\right)} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot y\right) \cdot y} \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{y \cdot \left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot y\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot y\right)} \]

       7. associate-*r* N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\left({t}^{2} \cdot x\right) \cdot y\right)\right)} \]

       8. associate-*r* N/A

          \[\leadsto y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({t}^{2} \cdot \left(x \cdot y\right)\right)}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot y\right)\right)\right)} \]

       10. associate-*r* N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left({t}^{2} \cdot x\right) \cdot y\right)}\right) \]

       11. *-commutative N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot \left({t}^{2} \cdot x\right)\right)}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot \left({t}^{2} \cdot x\right)\right)}\right) \]

       13. *-commutative N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \left(y \cdot \color{blue}{\left(x \cdot {t}^{2}\right)}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \left(y \cdot \color{blue}{\left(x \cdot {t}^{2}\right)}\right)\right) \]

       15. unpow2 N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \left(y \cdot \left(x \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

       16. *-lowering-*.f64 32.7

           \[\leadsto y \cdot \left(0.5 \cdot \left(y \cdot \left(x \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

   14. Simplified 32.7%

       \[\leadsto \color{blue}{y \cdot \left(0.5 \cdot \left(y \cdot \left(x \cdot \left(t \cdot t\right)\right)\right)\right)} \]

   if -9.9999999999999993e44 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5e13

   1. Initial program 93.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 78.8

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 78.8%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + -1 \cdot a}, 1\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot b, -1 \cdot a\right)}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{b \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{b \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      9. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \left(a \cdot a\right), \color{blue}{\mathsf{neg}\left(a\right)}\right), 1\right) \]

      10. neg-lowering-neg.f64 81.7

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), \color{blue}{-a}\right), 1\right) \]

   8. Simplified 81.7%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)} \]

   if 1.99999999999999996e150 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 95.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 60.4

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 60.4%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + t \cdot \left(-1 \cdot y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(t \cdot \left(-1 \cdot y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot y + \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) + -1 \cdot y}, 1\right) \]

      4. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}, 1\right) \]

      5. unsub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) - y}, 1\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right) - y}, 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \color{blue}{\frac{1}{2} \cdot \left(t \cdot {y}^{2}\right)} - y, 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \frac{1}{2} \cdot \color{blue}{\left(t \cdot {y}^{2}\right)} - y, 1\right) \]

      9. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(t, \frac{1}{2} \cdot \left(t \cdot \color{blue}{\left(y \cdot y\right)}\right) - y, 1\right) \]

      10. *-lowering-*.f64 73.7

          \[\leadsto x \cdot \mathsf{fma}\left(t, 0.5 \cdot \left(t \cdot \color{blue}{\left(y \cdot y\right)}\right) - y, 1\right) \]

   8. Simplified 73.7%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, 0.5 \cdot \left(t \cdot \left(y \cdot y\right)\right) - y, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 37.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -20000000000000:\\ \;\;\;\;b \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;t\_1 \leq 50000000000000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(z + b\right), -x, x\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+303}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, -b, \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -20000000000000.0)
     (* b (* x (- a)))
     (if (<= t_1 50000000000000.0)
       (fma (* a (+ z b)) (- x) x)
       (if (<= t_1 4e+303)
         (* a (fma x (- b) (/ x a)))
         (* x (- 1.0 (* y t))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -20000000000000.0) {
		tmp = b * (x * -a);
	} else if (t_1 <= 50000000000000.0) {
		tmp = fma((a * (z + b)), -x, x);
	} else if (t_1 <= 4e+303) {
		tmp = a * fma(x, -b, (x / a));
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -20000000000000.0)
		tmp = Float64(b * Float64(x * Float64(-a)));
	elseif (t_1 <= 50000000000000.0)
		tmp = fma(Float64(a * Float64(z + b)), Float64(-x), x);
	elseif (t_1 <= 4e+303)
		tmp = Float64(a * fma(x, Float64(-b), Float64(x / a)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000000000.0], N[(b * N[(x * (-a)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 50000000000000.0], N[(N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision] * (-x) + x), $MachinePrecision], If[LessEqual[t$95$1, 4e+303], N[(a * N[(x * (-b) + N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e13

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 49.3

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 49.3%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)\right)} + x \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot b\right) \cdot x}\right)\right) + x \]

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

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} + x \]

      5. mul-1-neg N/A

         \[\leadsto \left(a \cdot b\right) \cdot \color{blue}{\left(-1 \cdot x\right)} + x \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot b, -1 \cdot x, x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot b}, -1 \cdot x, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{\mathsf{neg}\left(x\right)}, x\right) \]

      9. neg-lowering-neg.f64 3.0

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{-x}, x\right) \]

   8. Simplified 3.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot b, -x, x\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(b \cdot x\right) \cdot a}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(x \cdot b\right)} \cdot a\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \left(b \cdot a\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{\left(a \cdot b\right)}\right) \]

       6. distribute-rgt-neg-out N/A

          \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(a \cdot b\right)\right)} \]

       7. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \left(a \cdot b\right)\right)} \]

       9. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)} \]

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

           \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(\mathsf{neg}\left(b\right)\right)\right)} \]

       11. mul-1-neg N/A

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-1 \cdot b\right)}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b\right)\right)} \]

       13. mul-1-neg N/A

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

       14. neg-lowering-neg.f64 18.4

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-b\right)}\right) \]

   11. Simplified 18.4%

       \[\leadsto \color{blue}{x \cdot \left(a \cdot \left(-b\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(x \cdot a\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \cdot a\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \cdot a\right)} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

       4. neg-lowering-neg.f64 20.4

          \[\leadsto \left(x \cdot a\right) \cdot \color{blue}{\left(-b\right)} \]

   13. Applied egg-rr 20.4%

       \[\leadsto \color{blue}{\left(x \cdot a\right) \cdot \left(-b\right)} \]

   if -2e13 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5e13

   1. Initial program 93.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\log \left(1 - z\right) - b\right)}} \]

      3. sub-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right)} \]

      4. accelerator-lowering-log1p.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      5. neg-lowering-neg.f64 93.9

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Simplified 93.9%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. neg-lowering-neg.f64 93.9

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Simplified 93.9%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right) + x} \]

       2. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)\right)} + x \]

       3. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(a \cdot \color{blue}{\left(\left(b + z\right) \cdot x\right)}\right)\right) + x \]

       4. associate-*r* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot \left(b + z\right)\right) \cdot x}\right)\right) + x \]

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

          \[\leadsto \color{blue}{\left(a \cdot \left(b + z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} + x \]

       6. mul-1-neg N/A

          \[\leadsto \left(a \cdot \left(b + z\right)\right) \cdot \color{blue}{\left(-1 \cdot x\right)} + x \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \left(b + z\right), -1 \cdot x, x\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot \left(b + z\right)}, -1 \cdot x, x\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot \color{blue}{\left(b + z\right)}, -1 \cdot x, x\right) \]

       10. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(a \cdot \left(b + z\right), \color{blue}{\mathsf{neg}\left(x\right)}, x\right) \]

       11. neg-lowering-neg.f64 84.5

           \[\leadsto \mathsf{fma}\left(a \cdot \left(b + z\right), \color{blue}{-x}, x\right) \]

   11. Simplified 84.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \left(b + z\right), -x, x\right)} \]

   if 5e13 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4e303

   1. Initial program 93.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 40.1

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 40.1%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)\right)} + x \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot b\right) \cdot x}\right)\right) + x \]

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

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} + x \]

      5. mul-1-neg N/A

         \[\leadsto \left(a \cdot b\right) \cdot \color{blue}{\left(-1 \cdot x\right)} + x \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot b, -1 \cdot x, x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot b}, -1 \cdot x, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{\mathsf{neg}\left(x\right)}, x\right) \]

      9. neg-lowering-neg.f64 11.3

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{-x}, x\right) \]

   8. Simplified 11.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot b, -x, x\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(b \cdot x\right) + \frac{x}{a}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(b \cdot x\right) + \frac{x}{a}\right)} \]

       2. mul-1-neg N/A

          \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot x\right)\right)} + \frac{x}{a}\right) \]

       3. *-commutative N/A

          \[\leadsto a \cdot \left(\left(\mathsf{neg}\left(\color{blue}{x \cdot b}\right)\right) + \frac{x}{a}\right) \]

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

          \[\leadsto a \cdot \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(b\right)\right)} + \frac{x}{a}\right) \]

       5. mul-1-neg N/A

          \[\leadsto a \cdot \left(x \cdot \color{blue}{\left(-1 \cdot b\right)} + \frac{x}{a}\right) \]

       6. accelerator-lowering-fma.f64 N/A

          \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(x, -1 \cdot b, \frac{x}{a}\right)} \]

       7. mul-1-neg N/A

          \[\leadsto a \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{neg}\left(b\right)}, \frac{x}{a}\right) \]

       8. neg-lowering-neg.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{neg}\left(b\right)}, \frac{x}{a}\right) \]

       9. /-lowering-/.f64 20.7

          \[\leadsto a \cdot \mathsf{fma}\left(x, -b, \color{blue}{\frac{x}{a}}\right) \]

   11. Simplified 20.7%

       \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(x, -b, \frac{x}{a}\right)} \]

   if 4e303 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 90.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 66.0

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 66.0%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(t \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(t \cdot y\right)\right)}\right) \]

      2. unsub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(1 - \color{blue}{y \cdot t}\right) \]

      5. *-lowering-*.f64 61.5

         \[\leadsto x \cdot \left(1 - \color{blue}{y \cdot t}\right) \]

   8. Simplified 61.5%

      \[\leadsto x \cdot \color{blue}{\left(1 - y \cdot t\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000000:\\ \;\;\;\;b \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 50000000000000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(z + b\right), -x, x\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 4 \cdot 10^{+303}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, -b, \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+45}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \left(y \cdot \left(x \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 50000000000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(t \cdot t\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -1e+45)
     (* y (* 0.5 (* y (* x (* t t)))))
     (if (<= t_1 50000000000000.0)
       (* x (fma b (fma 0.5 (* b (* a a)) (- a)) 1.0))
       (* x (* 0.5 (* y (* y (* t t)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -1e+45) {
		tmp = y * (0.5 * (y * (x * (t * t))));
	} else if (t_1 <= 50000000000000.0) {
		tmp = x * fma(b, fma(0.5, (b * (a * a)), -a), 1.0);
	} else {
		tmp = x * (0.5 * (y * (y * (t * t))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -1e+45)
		tmp = Float64(y * Float64(0.5 * Float64(y * Float64(x * Float64(t * t)))));
	elseif (t_1 <= 50000000000000.0)
		tmp = Float64(x * fma(b, fma(0.5, Float64(b * Float64(a * a)), Float64(-a)), 1.0));
	else
		tmp = Float64(x * Float64(0.5 * Float64(y * Float64(y * Float64(t * t)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+45], N[(y * N[(0.5 * N[(y * N[(x * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 50000000000000.0], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.5 * N[(y * N[(y * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -9.9999999999999993e44

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 53.7

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 53.7%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-lowering-neg.f64 2.5

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{-t}\right), 1\right) \]

   8. Simplified 2.5%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{t \cdot \left(\frac{1}{2} \cdot \left(t \cdot y\right) - 1\right)}, 1\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{t \cdot \left(\frac{1}{2} \cdot \left(t \cdot y\right) - 1\right)}, 1\right) \]

       2. sub-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, t \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(t \cdot y\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}, 1\right) \]

       3. metadata-eval N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, t \cdot \left(\frac{1}{2} \cdot \left(t \cdot y\right) + \color{blue}{-1}\right), 1\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, t \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, t \cdot y, -1\right)}, 1\right) \]

       5. *-lowering-*.f64 2.5

          \[\leadsto x \cdot \mathsf{fma}\left(y, t \cdot \mathsf{fma}\left(0.5, \color{blue}{t \cdot y}, -1\right), 1\right) \]

   11. Simplified 2.5%

       \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{t \cdot \mathsf{fma}\left(0.5, t \cdot y, -1\right)}, 1\right) \]

   12. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left({t}^{2} \cdot x\right) \cdot {y}^{2}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot {y}^{2}} \]

       3. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot \color{blue}{\left(y \cdot y\right)} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot y\right) \cdot y} \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{y \cdot \left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot y\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot y\right)} \]

       7. associate-*r* N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\left({t}^{2} \cdot x\right) \cdot y\right)\right)} \]

       8. associate-*r* N/A

          \[\leadsto y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({t}^{2} \cdot \left(x \cdot y\right)\right)}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot y\right)\right)\right)} \]

       10. associate-*r* N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left({t}^{2} \cdot x\right) \cdot y\right)}\right) \]

       11. *-commutative N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot \left({t}^{2} \cdot x\right)\right)}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot \left({t}^{2} \cdot x\right)\right)}\right) \]

       13. *-commutative N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \left(y \cdot \color{blue}{\left(x \cdot {t}^{2}\right)}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \left(y \cdot \color{blue}{\left(x \cdot {t}^{2}\right)}\right)\right) \]

       15. unpow2 N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \left(y \cdot \left(x \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

       16. *-lowering-*.f64 31.5

           \[\leadsto y \cdot \left(0.5 \cdot \left(y \cdot \left(x \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

   14. Simplified 31.5%

       \[\leadsto \color{blue}{y \cdot \left(0.5 \cdot \left(y \cdot \left(x \cdot \left(t \cdot t\right)\right)\right)\right)} \]

   if -9.9999999999999993e44 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5e13

   1. Initial program 93.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 78.8

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 78.8%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right), 1\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + -1 \cdot a}, 1\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot b, -1 \cdot a\right)}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{b \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{b \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      9. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \left(a \cdot a\right), \color{blue}{\mathsf{neg}\left(a\right)}\right), 1\right) \]

      10. neg-lowering-neg.f64 81.7

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), \color{blue}{-a}\right), 1\right) \]

   8. Simplified 81.7%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)} \]

   if 5e13 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 92.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 52.0

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 52.0%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-lowering-neg.f64 57.9

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{-t}\right), 1\right) \]

   8. Simplified 57.9%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot {y}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({y}^{2} \cdot {t}^{2}\right)}\right) \]

       3. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot {t}^{2}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot \left(y \cdot {t}^{2}\right)\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(y \cdot \color{blue}{\left({t}^{2} \cdot y\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot \left({t}^{2} \cdot y\right)\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(y \cdot \color{blue}{\left({t}^{2} \cdot y\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(y \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot y\right)\right)\right) \]

       9. *-lowering-*.f64 60.1

          \[\leadsto x \cdot \left(0.5 \cdot \left(y \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot y\right)\right)\right) \]

   11. Simplified 60.1%

       \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(y \cdot \left(\left(t \cdot t\right) \cdot y\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -1 \cdot 10^{+45}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \left(y \cdot \left(x \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 50000000000000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(t \cdot t\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+45}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \left(y \cdot \left(x \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 50000000000000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(z + b\right), -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(t \cdot t\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -1e+45)
     (* y (* 0.5 (* y (* x (* t t)))))
     (if (<= t_1 50000000000000.0)
       (fma (* a (+ z b)) (- x) x)
       (* x (* 0.5 (* y (* y (* t t)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -1e+45) {
		tmp = y * (0.5 * (y * (x * (t * t))));
	} else if (t_1 <= 50000000000000.0) {
		tmp = fma((a * (z + b)), -x, x);
	} else {
		tmp = x * (0.5 * (y * (y * (t * t))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -1e+45)
		tmp = Float64(y * Float64(0.5 * Float64(y * Float64(x * Float64(t * t)))));
	elseif (t_1 <= 50000000000000.0)
		tmp = fma(Float64(a * Float64(z + b)), Float64(-x), x);
	else
		tmp = Float64(x * Float64(0.5 * Float64(y * Float64(y * Float64(t * t)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+45], N[(y * N[(0.5 * N[(y * N[(x * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 50000000000000.0], N[(N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision] * (-x) + x), $MachinePrecision], N[(x * N[(0.5 * N[(y * N[(y * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -9.9999999999999993e44

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 53.7

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 53.7%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-lowering-neg.f64 2.5

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{-t}\right), 1\right) \]

   8. Simplified 2.5%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{t \cdot \left(\frac{1}{2} \cdot \left(t \cdot y\right) - 1\right)}, 1\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{t \cdot \left(\frac{1}{2} \cdot \left(t \cdot y\right) - 1\right)}, 1\right) \]

       2. sub-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, t \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(t \cdot y\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}, 1\right) \]

       3. metadata-eval N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, t \cdot \left(\frac{1}{2} \cdot \left(t \cdot y\right) + \color{blue}{-1}\right), 1\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, t \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, t \cdot y, -1\right)}, 1\right) \]

       5. *-lowering-*.f64 2.5

          \[\leadsto x \cdot \mathsf{fma}\left(y, t \cdot \mathsf{fma}\left(0.5, \color{blue}{t \cdot y}, -1\right), 1\right) \]

   11. Simplified 2.5%

       \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{t \cdot \mathsf{fma}\left(0.5, t \cdot y, -1\right)}, 1\right) \]

   12. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left({t}^{2} \cdot x\right) \cdot {y}^{2}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot {y}^{2}} \]

       3. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot \color{blue}{\left(y \cdot y\right)} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot y\right) \cdot y} \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{y \cdot \left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot y\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot \left(\left(\frac{1}{2} \cdot \left({t}^{2} \cdot x\right)\right) \cdot y\right)} \]

       7. associate-*r* N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\left({t}^{2} \cdot x\right) \cdot y\right)\right)} \]

       8. associate-*r* N/A

          \[\leadsto y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({t}^{2} \cdot \left(x \cdot y\right)\right)}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(x \cdot y\right)\right)\right)} \]

       10. associate-*r* N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left({t}^{2} \cdot x\right) \cdot y\right)}\right) \]

       11. *-commutative N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot \left({t}^{2} \cdot x\right)\right)}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot \left({t}^{2} \cdot x\right)\right)}\right) \]

       13. *-commutative N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \left(y \cdot \color{blue}{\left(x \cdot {t}^{2}\right)}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \left(y \cdot \color{blue}{\left(x \cdot {t}^{2}\right)}\right)\right) \]

       15. unpow2 N/A

           \[\leadsto y \cdot \left(\frac{1}{2} \cdot \left(y \cdot \left(x \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

       16. *-lowering-*.f64 31.5

           \[\leadsto y \cdot \left(0.5 \cdot \left(y \cdot \left(x \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

   14. Simplified 31.5%

       \[\leadsto \color{blue}{y \cdot \left(0.5 \cdot \left(y \cdot \left(x \cdot \left(t \cdot t\right)\right)\right)\right)} \]

   if -9.9999999999999993e44 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5e13

   1. Initial program 93.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\log \left(1 - z\right) - b\right)}} \]

      3. sub-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right)} \]

      4. accelerator-lowering-log1p.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      5. neg-lowering-neg.f64 86.6

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Simplified 86.6%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. neg-lowering-neg.f64 86.6

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Simplified 86.6%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right) + x} \]

       2. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)\right)} + x \]

       3. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(a \cdot \color{blue}{\left(\left(b + z\right) \cdot x\right)}\right)\right) + x \]

       4. associate-*r* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot \left(b + z\right)\right) \cdot x}\right)\right) + x \]

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

          \[\leadsto \color{blue}{\left(a \cdot \left(b + z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} + x \]

       6. mul-1-neg N/A

          \[\leadsto \left(a \cdot \left(b + z\right)\right) \cdot \color{blue}{\left(-1 \cdot x\right)} + x \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \left(b + z\right), -1 \cdot x, x\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot \left(b + z\right)}, -1 \cdot x, x\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot \color{blue}{\left(b + z\right)}, -1 \cdot x, x\right) \]

       10. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(a \cdot \left(b + z\right), \color{blue}{\mathsf{neg}\left(x\right)}, x\right) \]

       11. neg-lowering-neg.f64 77.9

           \[\leadsto \mathsf{fma}\left(a \cdot \left(b + z\right), \color{blue}{-x}, x\right) \]

   11. Simplified 77.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \left(b + z\right), -x, x\right)} \]

   if 5e13 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 92.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 52.0

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 52.0%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-lowering-neg.f64 57.9

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{-t}\right), 1\right) \]

   8. Simplified 57.9%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot {y}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({y}^{2} \cdot {t}^{2}\right)}\right) \]

       3. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot {t}^{2}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot \left(y \cdot {t}^{2}\right)\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(y \cdot \color{blue}{\left({t}^{2} \cdot y\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot \left({t}^{2} \cdot y\right)\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(y \cdot \color{blue}{\left({t}^{2} \cdot y\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(y \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot y\right)\right)\right) \]

       9. *-lowering-*.f64 60.1

          \[\leadsto x \cdot \left(0.5 \cdot \left(y \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot y\right)\right)\right) \]

   11. Simplified 60.1%

       \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(y \cdot \left(\left(t \cdot t\right) \cdot y\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -1 \cdot 10^{+45}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \left(y \cdot \left(x \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 50000000000000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(z + b\right), -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(t \cdot t\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(t \cdot t\right)\right)\right)\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 50000000000000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(z + b\right), -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (* 0.5 (* y (* y (* t t))))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -1e+45)
     t_1
     (if (<= t_2 50000000000000.0) (fma (* a (+ z b)) (- x) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (0.5 * (y * (y * (t * t))));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -1e+45) {
		tmp = t_1;
	} else if (t_2 <= 50000000000000.0) {
		tmp = fma((a * (z + b)), -x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64(0.5 * Float64(y * Float64(y * Float64(t * t)))))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -1e+45)
		tmp = t_1;
	elseif (t_2 <= 50000000000000.0)
		tmp = fma(Float64(a * Float64(z + b)), Float64(-x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(0.5 * N[(y * N[(y * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+45], t$95$1, If[LessEqual[t$95$2, 50000000000000.0], N[(N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision] * (-x) + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -9.9999999999999993e44 or 5e13 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 95.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 52.8

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 52.8%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-lowering-neg.f64 31.7

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{-t}\right), 1\right) \]

   8. Simplified 31.7%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot {y}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({y}^{2} \cdot {t}^{2}\right)}\right) \]

       3. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot {t}^{2}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot \left(y \cdot {t}^{2}\right)\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(y \cdot \color{blue}{\left({t}^{2} \cdot y\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot \left({t}^{2} \cdot y\right)\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(y \cdot \color{blue}{\left({t}^{2} \cdot y\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(y \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot y\right)\right)\right) \]

       9. *-lowering-*.f64 46.2

          \[\leadsto x \cdot \left(0.5 \cdot \left(y \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot y\right)\right)\right) \]

   11. Simplified 46.2%

       \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(y \cdot \left(\left(t \cdot t\right) \cdot y\right)\right)\right)} \]

   if -9.9999999999999993e44 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5e13

   1. Initial program 93.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\log \left(1 - z\right) - b\right)}} \]

      3. sub-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right)} \]

      4. accelerator-lowering-log1p.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      5. neg-lowering-neg.f64 86.6

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Simplified 86.6%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. neg-lowering-neg.f64 86.6

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Simplified 86.6%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(x \cdot \left(b + z\right)\right)\right) + x} \]

       2. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(x \cdot \left(b + z\right)\right)\right)\right)} + x \]

       3. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(a \cdot \color{blue}{\left(\left(b + z\right) \cdot x\right)}\right)\right) + x \]

       4. associate-*r* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot \left(b + z\right)\right) \cdot x}\right)\right) + x \]

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

          \[\leadsto \color{blue}{\left(a \cdot \left(b + z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} + x \]

       6. mul-1-neg N/A

          \[\leadsto \left(a \cdot \left(b + z\right)\right) \cdot \color{blue}{\left(-1 \cdot x\right)} + x \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \left(b + z\right), -1 \cdot x, x\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot \left(b + z\right)}, -1 \cdot x, x\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot \color{blue}{\left(b + z\right)}, -1 \cdot x, x\right) \]

       10. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(a \cdot \left(b + z\right), \color{blue}{\mathsf{neg}\left(x\right)}, x\right) \]

       11. neg-lowering-neg.f64 77.9

           \[\leadsto \mathsf{fma}\left(a \cdot \left(b + z\right), \color{blue}{-x}, x\right) \]

   11. Simplified 77.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \left(b + z\right), -x, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -1 \cdot 10^{+45}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 50000000000000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(z + b\right), -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(t \cdot t\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -20000000000000:\\ \;\;\;\;b \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+37}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -20000000000000.0)
     (* b (* x (- a)))
     (if (<= t_1 5e+37) x (* (* a b) (- x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -20000000000000.0) {
		tmp = b * (x * -a);
	} else if (t_1 <= 5e+37) {
		tmp = x;
	} else {
		tmp = (a * b) * -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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) :: t_1
    real(8) :: tmp
    t_1 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_1 <= (-20000000000000.0d0)) then
        tmp = b * (x * -a)
    else if (t_1 <= 5d+37) then
        tmp = x
    else
        tmp = (a * b) * -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -20000000000000.0) {
		tmp = b * (x * -a);
	} else if (t_1 <= 5e+37) {
		tmp = x;
	} else {
		tmp = (a * b) * -x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_1 <= -20000000000000.0:
		tmp = b * (x * -a)
	elif t_1 <= 5e+37:
		tmp = x
	else:
		tmp = (a * b) * -x
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -20000000000000.0)
		tmp = Float64(b * Float64(x * Float64(-a)));
	elseif (t_1 <= 5e+37)
		tmp = x;
	else
		tmp = Float64(Float64(a * b) * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_1 <= -20000000000000.0)
		tmp = b * (x * -a);
	elseif (t_1 <= 5e+37)
		tmp = x;
	else
		tmp = (a * b) * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000000000.0], N[(b * N[(x * (-a)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+37], x, N[(N[(a * b), $MachinePrecision] * (-x)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e13

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 49.3

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 49.3%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)\right)} + x \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot b\right) \cdot x}\right)\right) + x \]

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

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} + x \]

      5. mul-1-neg N/A

         \[\leadsto \left(a \cdot b\right) \cdot \color{blue}{\left(-1 \cdot x\right)} + x \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot b, -1 \cdot x, x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot b}, -1 \cdot x, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{\mathsf{neg}\left(x\right)}, x\right) \]

      9. neg-lowering-neg.f64 3.0

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{-x}, x\right) \]

   8. Simplified 3.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot b, -x, x\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(b \cdot x\right) \cdot a}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(x \cdot b\right)} \cdot a\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \left(b \cdot a\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{\left(a \cdot b\right)}\right) \]

       6. distribute-rgt-neg-out N/A

          \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(a \cdot b\right)\right)} \]

       7. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \left(a \cdot b\right)\right)} \]

       9. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)} \]

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

           \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(\mathsf{neg}\left(b\right)\right)\right)} \]

       11. mul-1-neg N/A

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-1 \cdot b\right)}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b\right)\right)} \]

       13. mul-1-neg N/A

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

       14. neg-lowering-neg.f64 18.4

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-b\right)}\right) \]

   11. Simplified 18.4%

       \[\leadsto \color{blue}{x \cdot \left(a \cdot \left(-b\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(x \cdot a\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \cdot a\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \cdot a\right)} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

       4. neg-lowering-neg.f64 20.4

          \[\leadsto \left(x \cdot a\right) \cdot \color{blue}{\left(-b\right)} \]

   13. Applied egg-rr 20.4%

       \[\leadsto \color{blue}{\left(x \cdot a\right) \cdot \left(-b\right)} \]

   if -2e13 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.99999999999999989e37

   1. Initial program 91.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. log-lowering-log.f64 84.4

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 84.4%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} \]
   7. Step-by-step derivation

   8. Simplified 75.8%

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

   if 4.99999999999999989e37 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 93.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 48.8

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 48.8%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)\right)} + x \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot b\right) \cdot x}\right)\right) + x \]

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

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} + x \]

      5. mul-1-neg N/A

         \[\leadsto \left(a \cdot b\right) \cdot \color{blue}{\left(-1 \cdot x\right)} + x \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot b, -1 \cdot x, x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot b}, -1 \cdot x, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{\mathsf{neg}\left(x\right)}, x\right) \]

      9. neg-lowering-neg.f64 28.4

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{-x}, x\right) \]

   8. Simplified 28.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot b, -x, x\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(b \cdot x\right) \cdot a}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(x \cdot b\right)} \cdot a\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \left(b \cdot a\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{\left(a \cdot b\right)}\right) \]

       6. distribute-rgt-neg-out N/A

          \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(a \cdot b\right)\right)} \]

       7. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \left(a \cdot b\right)\right)} \]

       9. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)} \]

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

           \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(\mathsf{neg}\left(b\right)\right)\right)} \]

       11. mul-1-neg N/A

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-1 \cdot b\right)}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b\right)\right)} \]

       13. mul-1-neg N/A

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

       14. neg-lowering-neg.f64 28.0

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-b\right)}\right) \]

   11. Simplified 28.0%

       \[\leadsto \color{blue}{x \cdot \left(a \cdot \left(-b\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000000:\\ \;\;\;\;b \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+37}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 32.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot b\right) \cdot \left(-x\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -20000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+37}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* a b) (- x)))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -20000000000000.0) t_1 (if (<= t_2 5e+37) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) * -x;
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -20000000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+37) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a * b) * -x
    t_2 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_2 <= (-20000000000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 5d+37) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) * -x;
	double t_2 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -20000000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+37) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a * b) * -x
	t_2 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_2 <= -20000000000000.0:
		tmp = t_1
	elif t_2 <= 5e+37:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * b) * Float64(-x))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -20000000000000.0)
		tmp = t_1;
	elseif (t_2 <= 5e+37)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * b) * -x;
	t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_2 <= -20000000000000.0)
		tmp = t_1;
	elseif (t_2 <= 5e+37)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] * (-x)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -20000000000000.0], t$95$1, If[LessEqual[t$95$2, 5e+37], x, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e13 or 4.99999999999999989e37 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 96.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 49.1

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 49.1%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)\right)} + x \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot b\right) \cdot x}\right)\right) + x \]

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

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} + x \]

      5. mul-1-neg N/A

         \[\leadsto \left(a \cdot b\right) \cdot \color{blue}{\left(-1 \cdot x\right)} + x \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot b, -1 \cdot x, x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot b}, -1 \cdot x, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{\mathsf{neg}\left(x\right)}, x\right) \]

      9. neg-lowering-neg.f64 15.8

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{-x}, x\right) \]

   8. Simplified 15.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot b, -x, x\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(b \cdot x\right) \cdot a}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(x \cdot b\right)} \cdot a\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \left(b \cdot a\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{\left(a \cdot b\right)}\right) \]

       6. distribute-rgt-neg-out N/A

          \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(a \cdot b\right)\right)} \]

       7. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \left(a \cdot b\right)\right)} \]

       9. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)} \]

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

           \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(\mathsf{neg}\left(b\right)\right)\right)} \]

       11. mul-1-neg N/A

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-1 \cdot b\right)}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b\right)\right)} \]

       13. mul-1-neg N/A

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

       14. neg-lowering-neg.f64 23.2

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-b\right)}\right) \]

   11. Simplified 23.2%

       \[\leadsto \color{blue}{x \cdot \left(a \cdot \left(-b\right)\right)} \]

   if -2e13 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.99999999999999989e37

   1. Initial program 91.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. log-lowering-log.f64 84.4

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 84.4%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} \]
   7. Step-by-step derivation

   8. Simplified 75.8%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -20000000000000:\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(-x\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+37}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0:\\ \;\;\;\;b \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))) 0.0)
   (* b (* x (- a)))
   (* x (- 1.0 (* y t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b)))) <= 0.0) {
		tmp = b * (x * -a);
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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) :: tmp
    if (exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b)))) <= 0.0d0) then
        tmp = b * (x * -a)
    else
        tmp = x * (1.0d0 - (y * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b)))) <= 0.0) {
		tmp = b * (x * -a);
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b)))) <= 0.0:
		tmp = b * (x * -a)
	else:
		tmp = x * (1.0 - (y * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))) <= 0.0)
		tmp = Float64(b * Float64(x * Float64(-a)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b)))) <= 0.0)
		tmp = b * (x * -a);
	else
		tmp = x * (1.0 - (y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(b * N[(x * (-a)), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))) < 0.0

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 49.3

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 49.3%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)\right)} + x \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot b\right) \cdot x}\right)\right) + x \]

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

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} + x \]

      5. mul-1-neg N/A

         \[\leadsto \left(a \cdot b\right) \cdot \color{blue}{\left(-1 \cdot x\right)} + x \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot b, -1 \cdot x, x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot b}, -1 \cdot x, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{\mathsf{neg}\left(x\right)}, x\right) \]

      9. neg-lowering-neg.f64 3.0

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{-x}, x\right) \]

   8. Simplified 3.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot b, -x, x\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(b \cdot x\right) \cdot a}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(x \cdot b\right)} \cdot a\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \left(b \cdot a\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{\left(a \cdot b\right)}\right) \]

       6. distribute-rgt-neg-out N/A

          \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(a \cdot b\right)\right)} \]

       7. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \left(a \cdot b\right)\right)} \]

       9. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)} \]

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

           \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(\mathsf{neg}\left(b\right)\right)\right)} \]

       11. mul-1-neg N/A

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-1 \cdot b\right)}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b\right)\right)} \]

       13. mul-1-neg N/A

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

       14. neg-lowering-neg.f64 18.4

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-b\right)}\right) \]

   11. Simplified 18.4%

       \[\leadsto \color{blue}{x \cdot \left(a \cdot \left(-b\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(x \cdot a\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \cdot a\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \cdot a\right)} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

       4. neg-lowering-neg.f64 20.4

          \[\leadsto \left(x \cdot a\right) \cdot \color{blue}{\left(-b\right)} \]

   13. Applied egg-rr 20.4%

       \[\leadsto \color{blue}{\left(x \cdot a\right) \cdot \left(-b\right)} \]

   if 0.0 < (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))))

   1. Initial program 92.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 63.6

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 63.6%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(t \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(t \cdot y\right)\right)}\right) \]

      2. unsub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - t \cdot y\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(1 - \color{blue}{y \cdot t}\right) \]

      5. *-lowering-*.f64 48.3

         \[\leadsto x \cdot \left(1 - \color{blue}{y \cdot t}\right) \]

   8. Simplified 48.3%

      \[\leadsto x \cdot \color{blue}{\left(1 - y \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0:\\ \;\;\;\;b \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 33.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0:\\ \;\;\;\;b \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))) 0.0)
   (* b (* x (- a)))
   (* x (- 1.0 (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b)))) <= 0.0) {
		tmp = b * (x * -a);
	} else {
		tmp = x * (1.0 - (a * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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) :: tmp
    if (exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b)))) <= 0.0d0) then
        tmp = b * (x * -a)
    else
        tmp = x * (1.0d0 - (a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b)))) <= 0.0) {
		tmp = b * (x * -a);
	} else {
		tmp = x * (1.0 - (a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b)))) <= 0.0:
		tmp = b * (x * -a)
	else:
		tmp = x * (1.0 - (a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))) <= 0.0)
		tmp = Float64(b * Float64(x * Float64(-a)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b)))) <= 0.0)
		tmp = b * (x * -a);
	else
		tmp = x * (1.0 - (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(b * N[(x * (-a)), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))) < 0.0

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 49.3

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 49.3%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + -1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right) + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)\right)} + x \]

      3. associate-*r* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot b\right) \cdot x}\right)\right) + x \]

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

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} + x \]

      5. mul-1-neg N/A

         \[\leadsto \left(a \cdot b\right) \cdot \color{blue}{\left(-1 \cdot x\right)} + x \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot b, -1 \cdot x, x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot b}, -1 \cdot x, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{\mathsf{neg}\left(x\right)}, x\right) \]

      9. neg-lowering-neg.f64 3.0

         \[\leadsto \mathsf{fma}\left(a \cdot b, \color{blue}{-x}, x\right) \]

   8. Simplified 3.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot b, -x, x\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(b \cdot x\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(a \cdot \left(b \cdot x\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(b \cdot x\right) \cdot a}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(x \cdot b\right)} \cdot a\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \left(b \cdot a\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{\left(a \cdot b\right)}\right) \]

       6. distribute-rgt-neg-out N/A

          \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(a \cdot b\right)\right)} \]

       7. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(a \cdot b\right)\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \left(a \cdot b\right)\right)} \]

       9. mul-1-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)} \]

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

           \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(\mathsf{neg}\left(b\right)\right)\right)} \]

       11. mul-1-neg N/A

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-1 \cdot b\right)}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(-1 \cdot b\right)\right)} \]

       13. mul-1-neg N/A

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

       14. neg-lowering-neg.f64 18.4

           \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-b\right)}\right) \]

   11. Simplified 18.4%

       \[\leadsto \color{blue}{x \cdot \left(a \cdot \left(-b\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(x \cdot a\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \cdot a\right) \cdot \left(\mathsf{neg}\left(b\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(x \cdot a\right)} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

       4. neg-lowering-neg.f64 20.4

          \[\leadsto \left(x \cdot a\right) \cdot \color{blue}{\left(-b\right)} \]

   13. Applied egg-rr 20.4%

       \[\leadsto \color{blue}{\left(x \cdot a\right) \cdot \left(-b\right)} \]

   if 0.0 < (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))))

   1. Initial program 92.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 61.8

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 61.8%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + -1 \cdot \left(a \cdot b\right)\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right) \]

      2. unsub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]

      4. *-lowering-*.f64 46.9

         \[\leadsto x \cdot \left(1 - \color{blue}{a \cdot b}\right) \]

   8. Simplified 46.9%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0:\\ \;\;\;\;b \cdot \left(x \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{if}\;y \leq -1.26:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-37}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- (log z) t))))))
   (if (<= y -1.26)
     t_1
     (if (<= y 5.2e-37) (* x (exp (- (* a (+ z b))))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((y * (log(z) - t)));
	double tmp;
	if (y <= -1.26) {
		tmp = t_1;
	} else if (y <= 5.2e-37) {
		tmp = x * exp(-(a * (z + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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) :: t_1
    real(8) :: tmp
    t_1 = x * exp((y * (log(z) - t)))
    if (y <= (-1.26d0)) then
        tmp = t_1
    else if (y <= 5.2d-37) then
        tmp = x * exp(-(a * (z + b)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((y * (Math.log(z) - t)));
	double tmp;
	if (y <= -1.26) {
		tmp = t_1;
	} else if (y <= 5.2e-37) {
		tmp = x * Math.exp(-(a * (z + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((y * (math.log(z) - t)))
	tmp = 0
	if y <= -1.26:
		tmp = t_1
	elif y <= 5.2e-37:
		tmp = x * math.exp(-(a * (z + b)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(y * Float64(log(z) - t))))
	tmp = 0.0
	if (y <= -1.26)
		tmp = t_1;
	elseif (y <= 5.2e-37)
		tmp = Float64(x * exp(Float64(-Float64(a * Float64(z + b)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((y * (log(z) - t)));
	tmp = 0.0;
	if (y <= -1.26)
		tmp = t_1;
	elseif (y <= 5.2e-37)
		tmp = x * exp(-(a * (z + b)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.26], t$95$1, If[LessEqual[y, 5.2e-37], N[(x * N[Exp[(-N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.26000000000000001 or 5.19999999999999959e-37 < y

   1. Initial program 96.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. log-lowering-log.f64 88.7

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 88.7%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   if -1.26000000000000001 < y < 5.19999999999999959e-37

   1. Initial program 94.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\log \left(1 - z\right) - b\right)}} \]

      3. sub-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right)} \]

      4. accelerator-lowering-log1p.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      5. neg-lowering-neg.f64 88.0

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Simplified 88.0%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. neg-lowering-neg.f64 88.0

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Simplified 88.0%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.26:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-37}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+45}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-101}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {\left(e^{-1}\right)}^{\left(y \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.05e+45)
   (* x (exp (* t (- y))))
   (if (<= t 9.5e-101)
     (* x (exp (- (* a (+ z b)))))
     (* x (pow (exp -1.0) (* y t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.05e+45) {
		tmp = x * exp((t * -y));
	} else if (t <= 9.5e-101) {
		tmp = x * exp(-(a * (z + b)));
	} else {
		tmp = x * pow(exp(-1.0), (y * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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) :: tmp
    if (t <= (-1.05d+45)) then
        tmp = x * exp((t * -y))
    else if (t <= 9.5d-101) then
        tmp = x * exp(-(a * (z + b)))
    else
        tmp = x * (exp((-1.0d0)) ** (y * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.05e+45) {
		tmp = x * Math.exp((t * -y));
	} else if (t <= 9.5e-101) {
		tmp = x * Math.exp(-(a * (z + b)));
	} else {
		tmp = x * Math.pow(Math.exp(-1.0), (y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -1.05e+45:
		tmp = x * math.exp((t * -y))
	elif t <= 9.5e-101:
		tmp = x * math.exp(-(a * (z + b)))
	else:
		tmp = x * math.pow(math.exp(-1.0), (y * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1.05e+45)
		tmp = Float64(x * exp(Float64(t * Float64(-y))));
	elseif (t <= 9.5e-101)
		tmp = Float64(x * exp(Float64(-Float64(a * Float64(z + b)))));
	else
		tmp = Float64(x * (exp(-1.0) ^ Float64(y * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -1.05e+45)
		tmp = x * exp((t * -y));
	elseif (t <= 9.5e-101)
		tmp = x * exp(-(a * (z + b)));
	else
		tmp = x * (exp(-1.0) ^ (y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.05e+45], N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.5e-101], N[(x * N[Exp[(-N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[N[Exp[-1.0], $MachinePrecision], N[(y * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.04999999999999997e45

   1. Initial program 98.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 85.0

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 85.0%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   if -1.04999999999999997e45 < t < 9.49999999999999994e-101

   1. Initial program 94.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\log \left(1 - z\right) - b\right)}} \]

      3. sub-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right)} \]

      4. accelerator-lowering-log1p.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      5. neg-lowering-neg.f64 73.7

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Simplified 73.7%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. neg-lowering-neg.f64 73.7

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Simplified 73.7%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]

   if 9.49999999999999994e-101 < t

   1. Initial program 93.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 76.8

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 76.8%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
   6. Step-by-step derivation

      1. distribute-rgt-neg-out N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(y \cdot t\right)}} \]

      2. neg-mul-1 N/A

         \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(y \cdot t\right)}} \]

      3. exp-prod N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{-1}\right)}^{\left(y \cdot t\right)}} \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto x \cdot \color{blue}{{\left(e^{-1}\right)}^{\left(y \cdot t\right)}} \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto x \cdot {\color{blue}{\left(e^{-1}\right)}}^{\left(y \cdot t\right)} \]

      6. *-lowering-*.f64 76.8

         \[\leadsto x \cdot {\left(e^{-1}\right)}^{\color{blue}{\left(y \cdot t\right)}} \]

   7. Applied egg-rr 76.8%

      \[\leadsto x \cdot \color{blue}{{\left(e^{-1}\right)}^{\left(y \cdot t\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+45}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-101}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {\left(e^{-1}\right)}^{\left(y \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 74.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{if}\;t \leq -5.8 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-101}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* t (- y))))))
   (if (<= t -5.8e+44)
     t_1
     (if (<= t 9.5e-101) (* x (exp (- (* a (+ z b))))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((t * -y));
	double tmp;
	if (t <= -5.8e+44) {
		tmp = t_1;
	} else if (t <= 9.5e-101) {
		tmp = x * exp(-(a * (z + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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) :: t_1
    real(8) :: tmp
    t_1 = x * exp((t * -y))
    if (t <= (-5.8d+44)) then
        tmp = t_1
    else if (t <= 9.5d-101) then
        tmp = x * exp(-(a * (z + b)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((t * -y));
	double tmp;
	if (t <= -5.8e+44) {
		tmp = t_1;
	} else if (t <= 9.5e-101) {
		tmp = x * Math.exp(-(a * (z + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((t * -y))
	tmp = 0
	if t <= -5.8e+44:
		tmp = t_1
	elif t <= 9.5e-101:
		tmp = x * math.exp(-(a * (z + b)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(t * Float64(-y))))
	tmp = 0.0
	if (t <= -5.8e+44)
		tmp = t_1;
	elseif (t <= 9.5e-101)
		tmp = Float64(x * exp(Float64(-Float64(a * Float64(z + b)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((t * -y));
	tmp = 0.0;
	if (t <= -5.8e+44)
		tmp = t_1;
	elseif (t <= 9.5e-101)
		tmp = x * exp(-(a * (z + b)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.8e+44], t$95$1, If[LessEqual[t, 9.5e-101], N[(x * N[Exp[(-N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.8000000000000004e44 or 9.49999999999999994e-101 < t

   1. Initial program 96.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 80.5

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 80.5%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   if -5.8000000000000004e44 < t < 9.49999999999999994e-101

   1. Initial program 94.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\log \left(1 - z\right) - b\right)}} \]

      3. sub-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right)} \]

      4. accelerator-lowering-log1p.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      5. neg-lowering-neg.f64 73.7

         \[\leadsto x \cdot e^{a \cdot \left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right)} \]

   5. Simplified 73.7%

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\mathsf{log1p}\left(-z\right) - b\right)}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(-1 \cdot z - b\right)}} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - b\right)} \]

      3. neg-lowering-neg.f64 73.7

         \[\leadsto x \cdot e^{a \cdot \left(\color{blue}{\left(-z\right)} - b\right)} \]

   8. Simplified 73.7%

      \[\leadsto x \cdot e^{a \cdot \color{blue}{\left(\left(-z\right) - b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+44}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-101}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 71.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{if}\;t \leq -8.8 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-105}:\\ \;\;\;\;x \cdot e^{-a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* t (- y))))))
   (if (<= t -8.8e+89) t_1 (if (<= t 2.5e-105) (* x (exp (- (* a b)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((t * -y));
	double tmp;
	if (t <= -8.8e+89) {
		tmp = t_1;
	} else if (t <= 2.5e-105) {
		tmp = x * exp(-(a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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) :: t_1
    real(8) :: tmp
    t_1 = x * exp((t * -y))
    if (t <= (-8.8d+89)) then
        tmp = t_1
    else if (t <= 2.5d-105) then
        tmp = x * exp(-(a * b))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((t * -y));
	double tmp;
	if (t <= -8.8e+89) {
		tmp = t_1;
	} else if (t <= 2.5e-105) {
		tmp = x * Math.exp(-(a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((t * -y))
	tmp = 0
	if t <= -8.8e+89:
		tmp = t_1
	elif t <= 2.5e-105:
		tmp = x * math.exp(-(a * b))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(t * Float64(-y))))
	tmp = 0.0
	if (t <= -8.8e+89)
		tmp = t_1;
	elseif (t <= 2.5e-105)
		tmp = Float64(x * exp(Float64(-Float64(a * b))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((t * -y));
	tmp = 0.0;
	if (t <= -8.8e+89)
		tmp = t_1;
	elseif (t <= 2.5e-105)
		tmp = x * exp(-(a * b));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.8e+89], t$95$1, If[LessEqual[t, 2.5e-105], N[(x * N[Exp[(-N[(a * b), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.8000000000000001e89 or 2.49999999999999982e-105 < t

   1. Initial program 95.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 80.9

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 80.9%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   if -8.8000000000000001e89 < t < 2.49999999999999982e-105

   1. Initial program 94.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 67.7

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 67.7%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{+89}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-105}:\\ \;\;\;\;x \cdot e^{-a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 71.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -5.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+145}:\\ \;\;\;\;x \cdot e^{-a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -5.5) t_1 (if (<= y 1.1e+145) (* x (exp (- (* a b)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * pow(z, y);
	double tmp;
	if (y <= -5.5) {
		tmp = t_1;
	} else if (y <= 1.1e+145) {
		tmp = x * exp(-(a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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) :: t_1
    real(8) :: tmp
    t_1 = x * (z ** y)
    if (y <= (-5.5d0)) then
        tmp = t_1
    else if (y <= 1.1d+145) then
        tmp = x * exp(-(a * b))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.pow(z, y);
	double tmp;
	if (y <= -5.5) {
		tmp = t_1;
	} else if (y <= 1.1e+145) {
		tmp = x * Math.exp(-(a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.pow(z, y)
	tmp = 0
	if y <= -5.5:
		tmp = t_1
	elif y <= 1.1e+145:
		tmp = x * math.exp(-(a * b))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * (z ^ y))
	tmp = 0.0
	if (y <= -5.5)
		tmp = t_1;
	elseif (y <= 1.1e+145)
		tmp = Float64(x * exp(Float64(-Float64(a * b))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (z ^ y);
	tmp = 0.0;
	if (y <= -5.5)
		tmp = t_1;
	elseif (y <= 1.1e+145)
		tmp = x * exp(-(a * b));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.5], t$95$1, If[LessEqual[y, 1.1e+145], N[(x * N[Exp[(-N[(a * b), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5 or 1.10000000000000004e145 < y

   1. Initial program 96.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. log-lowering-log.f64 88.8

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 88.8%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

      2. pow-lowering-pow.f64 63.4

         \[\leadsto x \cdot \color{blue}{{z}^{y}} \]

   8. Simplified 63.4%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

   if -5.5 < y < 1.10000000000000004e145

   1. Initial program 94.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      3. *-lowering-*.f64 73.5

         \[\leadsto x \cdot e^{-\color{blue}{a \cdot b}} \]

   5. Simplified 73.5%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 58.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -3e+98)
   (* x (fma y (fma 0.5 (* y (* t t)) (- t)) 1.0))
   (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -3e+98) {
		tmp = x * fma(y, fma(0.5, (y * (t * t)), -t), 1.0);
	} else {
		tmp = x * pow(z, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -3e+98)
		tmp = Float64(x * fma(y, fma(0.5, Float64(y * Float64(t * t)), Float64(-t)), 1.0));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -3e+98], N[(x * N[(y * N[(0.5 * N[(y * N[(t * t), $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.0000000000000001e98

   1. Initial program 98.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(t \cdot y\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(t \cdot y\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{y \cdot t}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(t\right)\right)}} \]

      5. neg-lowering-neg.f64 86.0

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(-t\right)}} \]

   5. Simplified 86.0%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-lowering-neg.f64 51.1

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{-t}\right), 1\right) \]

   8. Simplified 51.1%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)} \]

   if -3.0000000000000001e98 < t

   1. Initial program 94.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

      3. log-lowering-log.f64 69.0

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

   5. Simplified 69.0%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot {z}^{y}} \]

      2. pow-lowering-pow.f64 63.4

         \[\leadsto x \cdot \color{blue}{{z}^{y}} \]

   8. Simplified 63.4%

      \[\leadsto \color{blue}{x \cdot {z}^{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 19.0% accurate, 328.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    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
    code = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

   2. --lowering--.f64 N/A

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log z - t\right)}} \]

   3. log-lowering-log.f64 72.9

      \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\log z} - t\right)} \]

5. Simplified 72.9%

   \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x} \]
7. Step-by-step derivation

8. Simplified 20.7%

   \[\leadsto \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))