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

Percentage Accurate: 96.7% → 96.7%

Time: 15.6s

Alternatives: 14

Speedup: 1.0×

• 41.1% 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: 96.7% 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: 96.7% 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 96.3%

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

Alternative 2: 57.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-x\right) \cdot \mathsf{fma}\left(a, b, \mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z \cdot a, a\right), -1\right)\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -2000:\\ \;\;\;\;-0.5 \cdot \left(a \cdot \left(x \cdot \left(z \cdot z\right)\right)\right)\\ \mathbf{elif}\;t\_2 \leq 5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, \left(-z\right) - b, \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- x) (fma a b (fma z (fma 0.5 (* z a) a) -1.0))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -2000.0)
     (* -0.5 (* a (* x (* z z))))
     (if (<= t_2 5.0)
       t_1
       (if (<= t_2 5e+295) (* a (fma x (- (- z) b) (/ x a))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -x * fma(a, b, fma(z, fma(0.5, (z * a), a), -1.0));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -2000.0) {
		tmp = -0.5 * (a * (x * (z * z)));
	} else if (t_2 <= 5.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+295) {
		tmp = a * fma(x, (-z - b), (x / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(-x) * fma(a, b, fma(z, fma(0.5, Float64(z * a), a), -1.0)))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -2000.0)
		tmp = Float64(-0.5 * Float64(a * Float64(x * Float64(z * z))));
	elseif (t_2 <= 5.0)
		tmp = t_1;
	elseif (t_2 <= 5e+295)
		tmp = Float64(a * fma(x, Float64(Float64(-z) - b), Float64(x / a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[((-x) * N[(a * b + N[(z * N[(0.5 * N[(z * a), $MachinePrecision] + a), $MachinePrecision] + -1.0), $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, -2000.0], N[(-0.5 * N[(a * N[(x * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5.0], t$95$1, If[LessEqual[t$95$2, 5e+295], N[(a * N[(x * N[((-z) - b), $MachinePrecision] + N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

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))) < -2e3

   1. Initial program 96.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 2.9%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 3.8%

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

   12. Taylor expanded in z around inf

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

   14. Applied rewrites 56.1%

       \[\leadsto -0.5 \cdot \left(a \cdot \left(x \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

   if -2e3 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5 or 4.99999999999999991e295 < (+.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 a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.2%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 57.9%

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

   12. Taylor expanded in x around -inf

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

   14. Applied rewrites 72.5%

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

   if 5 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.99999999999999991e295

   1. Initial program 97.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 10.9%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 22.4%

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

   12. Taylor expanded in z around 0

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

   14. Applied rewrites 22.4%

       \[\leadsto a \cdot \mathsf{fma}\left(x, \left(-z\right) - b, \frac{x}{a}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;-0.5 \cdot \left(a \cdot \left(x \cdot \left(z \cdot z\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5:\\ \;\;\;\;\left(-x\right) \cdot \mathsf{fma}\left(a, b, \mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z \cdot a, a\right), -1\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+295}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, \left(-z\right) - b, \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \mathsf{fma}\left(a, b, \mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z \cdot a, a\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -2000:\\ \;\;\;\;-0.5 \cdot \left(a \cdot \left(x \cdot \left(z \cdot z\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-23}:\\ \;\;\;\;x + \mathsf{fma}\left(z, x \cdot \left(-a\right), a \cdot \left(x \cdot \left(-b\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+305}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, \left(-z\right) - b, \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

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 -2000.0)
     (* -0.5 (* a (* x (* z z))))
     (if (<= t_1 1e-23)
       (+ x (fma z (* x (- a)) (* a (* x (- b)))))
       (if (<= t_1 1e+305)
         (* a (fma x (- (- z) b) (/ x a)))
         (- (* x (* a b))))))))

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 <= -2000.0) {
		tmp = -0.5 * (a * (x * (z * z)));
	} else if (t_1 <= 1e-23) {
		tmp = x + fma(z, (x * -a), (a * (x * -b)));
	} else if (t_1 <= 1e+305) {
		tmp = a * fma(x, (-z - b), (x / a));
	} else {
		tmp = -(x * (a * b));
	}
	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 <= -2000.0)
		tmp = Float64(-0.5 * Float64(a * Float64(x * Float64(z * z))));
	elseif (t_1 <= 1e-23)
		tmp = Float64(x + fma(z, Float64(x * Float64(-a)), Float64(a * Float64(x * Float64(-b)))));
	elseif (t_1 <= 1e+305)
		tmp = Float64(a * fma(x, Float64(Float64(-z) - b), Float64(x / a)));
	else
		tmp = Float64(-Float64(x * Float64(a * b)));
	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, -2000.0], N[(-0.5 * N[(a * N[(x * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-23], N[(x + N[(z * N[(x * (-a)), $MachinePrecision] + N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+305], N[(a * N[(x * N[((-z) - b), $MachinePrecision] + N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(x * N[(a * b), $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))) < -2e3

   1. Initial program 96.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 2.9%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 3.8%

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

   12. Taylor expanded in z around inf

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

   14. Applied rewrites 56.1%

       \[\leadsto -0.5 \cdot \left(a \cdot \left(x \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

   if -2e3 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.9999999999999996e-24

   1. Initial program 91.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 91.0%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 91.1%

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

   12. Taylor expanded in z around 0

       \[\leadsto x + \mathsf{fma}\left(z, -1 \cdot \left(a \cdot x\right), a \cdot \left(x \cdot \left(\mathsf{neg}\left(b\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 91.1%

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

   if 9.9999999999999996e-24 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.9999999999999994e304

   1. Initial program 97.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 12.4%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 23.1%

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

   12. Taylor expanded in z around 0

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

   14. Applied rewrites 23.1%

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

   if 9.9999999999999994e304 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 35.5%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 34.9%

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

   13. Applied rewrites 58.8%

       \[\leadsto \left(a \cdot \left(-b\right)\right) \cdot x \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;-0.5 \cdot \left(a \cdot \left(x \cdot \left(z \cdot z\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{-23}:\\ \;\;\;\;x + \mathsf{fma}\left(z, x \cdot \left(-a\right), a \cdot \left(x \cdot \left(-b\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{+305}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, \left(-z\right) - b, \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -2000:\\ \;\;\;\;-0.5 \cdot \left(a \cdot \left(x \cdot \left(z \cdot z\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(a, -\mathsf{fma}\left(x, b, x \cdot z\right), x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+305}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, \left(-z\right) - b, \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

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 -2000.0)
     (* -0.5 (* a (* x (* z z))))
     (if (<= t_1 1e-23)
       (fma a (- (fma x b (* x z))) x)
       (if (<= t_1 1e+305)
         (* a (fma x (- (- z) b) (/ x a)))
         (- (* x (* a b))))))))

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 <= -2000.0) {
		tmp = -0.5 * (a * (x * (z * z)));
	} else if (t_1 <= 1e-23) {
		tmp = fma(a, -fma(x, b, (x * z)), x);
	} else if (t_1 <= 1e+305) {
		tmp = a * fma(x, (-z - b), (x / a));
	} else {
		tmp = -(x * (a * b));
	}
	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 <= -2000.0)
		tmp = Float64(-0.5 * Float64(a * Float64(x * Float64(z * z))));
	elseif (t_1 <= 1e-23)
		tmp = fma(a, Float64(-fma(x, b, Float64(x * z))), x);
	elseif (t_1 <= 1e+305)
		tmp = Float64(a * fma(x, Float64(Float64(-z) - b), Float64(x / a)));
	else
		tmp = Float64(-Float64(x * Float64(a * b)));
	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, -2000.0], N[(-0.5 * N[(a * N[(x * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-23], N[(a * (-N[(x * b + N[(x * z), $MachinePrecision]), $MachinePrecision]) + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+305], N[(a * N[(x * N[((-z) - b), $MachinePrecision] + N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(x * N[(a * b), $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))) < -2e3

   1. Initial program 96.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 2.9%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 3.8%

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

   12. Taylor expanded in z around inf

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

   14. Applied rewrites 56.1%

       \[\leadsto -0.5 \cdot \left(a \cdot \left(x \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

   if -2e3 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.9999999999999996e-24

   1. Initial program 91.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 91.0%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 91.0%

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

   if 9.9999999999999996e-24 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.9999999999999994e304

   1. Initial program 97.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 12.4%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 23.1%

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

   12. Taylor expanded in z around 0

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

   14. Applied rewrites 23.1%

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

   if 9.9999999999999994e304 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 35.5%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 34.9%

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

   13. Applied rewrites 58.8%

       \[\leadsto \left(a \cdot \left(-b\right)\right) \cdot x \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;-0.5 \cdot \left(a \cdot \left(x \cdot \left(z \cdot z\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(a, -\mathsf{fma}\left(x, b, x \cdot z\right), x\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{+305}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, \left(-z\right) - b, \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -2000:\\ \;\;\;\;-0.5 \cdot \left(a \cdot \left(x \cdot \left(z \cdot z\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(a, -\mathsf{fma}\left(x, b, x \cdot z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

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 -2000.0)
     (* -0.5 (* a (* x (* z z))))
     (if (<= t_1 1e+305) (fma a (- (fma x b (* x z))) x) (- (* x (* a b)))))))

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 <= -2000.0) {
		tmp = -0.5 * (a * (x * (z * z)));
	} else if (t_1 <= 1e+305) {
		tmp = fma(a, -fma(x, b, (x * z)), x);
	} else {
		tmp = -(x * (a * b));
	}
	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 <= -2000.0)
		tmp = Float64(-0.5 * Float64(a * Float64(x * Float64(z * z))));
	elseif (t_1 <= 1e+305)
		tmp = fma(a, Float64(-fma(x, b, Float64(x * z))), x);
	else
		tmp = Float64(-Float64(x * Float64(a * b)));
	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, -2000.0], N[(-0.5 * N[(a * N[(x * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+305], N[(a * (-N[(x * b + N[(x * z), $MachinePrecision]), $MachinePrecision]) + x), $MachinePrecision], (-N[(x * N[(a * b), $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))) < -2e3

   1. Initial program 96.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 2.9%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 3.8%

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

   12. Taylor expanded in z around inf

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

   14. Applied rewrites 56.1%

       \[\leadsto -0.5 \cdot \left(a \cdot \left(x \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

   if -2e3 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.9999999999999994e304

   1. Initial program 95.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 39.7%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 39.7%

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

   if 9.9999999999999994e304 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 35.5%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 34.9%

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

   13. Applied rewrites 58.8%

       \[\leadsto \left(a \cdot \left(-b\right)\right) \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;-0.5 \cdot \left(a \cdot \left(x \cdot \left(z \cdot z\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(a, -\mathsf{fma}\left(x, b, x \cdot z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -2000:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(a, -\mathsf{fma}\left(x, b, x \cdot z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

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 -2000.0)
     (* a (* x (- b)))
     (if (<= t_1 1e+305) (fma a (- (fma x b (* x z))) x) (- (* x (* a b)))))))

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 <= -2000.0) {
		tmp = a * (x * -b);
	} else if (t_1 <= 1e+305) {
		tmp = fma(a, -fma(x, b, (x * z)), x);
	} else {
		tmp = -(x * (a * b));
	}
	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 <= -2000.0)
		tmp = Float64(a * Float64(x * Float64(-b)));
	elseif (t_1 <= 1e+305)
		tmp = fma(a, Float64(-fma(x, b, Float64(x * z))), x);
	else
		tmp = Float64(-Float64(x * Float64(a * b)));
	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, -2000.0], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+305], N[(a * (-N[(x * b + N[(x * z), $MachinePrecision]), $MachinePrecision]) + x), $MachinePrecision], (-N[(x * N[(a * b), $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))) < -2e3

   1. Initial program 96.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 2.9%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 18.2%

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

   if -2e3 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.9999999999999994e304

   1. Initial program 95.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 39.7%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 39.7%

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

   if 9.9999999999999994e304 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 97.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 35.5%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 34.9%

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

   13. Applied rewrites 58.8%

       \[\leadsto \left(a \cdot \left(-b\right)\right) \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(a, -\mathsf{fma}\left(x, b, x \cdot z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 33.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -2000:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(a, -x \cdot z, x\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \end{array} \end{array} \]

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 -2000.0)
     (* a (* x (- b)))
     (if (<= t_1 5e+74) (fma a (- (* x z)) x) (- (* x (* a b)))))))

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 <= -2000.0) {
		tmp = a * (x * -b);
	} else if (t_1 <= 5e+74) {
		tmp = fma(a, -(x * z), x);
	} else {
		tmp = -(x * (a * b));
	}
	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 <= -2000.0)
		tmp = Float64(a * Float64(x * Float64(-b)));
	elseif (t_1 <= 5e+74)
		tmp = fma(a, Float64(-Float64(x * z)), x);
	else
		tmp = Float64(-Float64(x * Float64(a * b)));
	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, -2000.0], N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+74], N[(a * (-N[(x * z), $MachinePrecision]) + x), $MachinePrecision], (-N[(x * N[(a * b), $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))) < -2e3

   1. Initial program 96.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 2.9%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 18.2%

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

   if -2e3 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.99999999999999963e74

   1. Initial program 92.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.1%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 61.1%

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

   12. Taylor expanded in b around 0

       \[\leadsto \mathsf{fma}\left(a, \mathsf{neg}\left(x \cdot z\right), x\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 61.3%

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

   if 4.99999999999999963e74 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 70.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 23.1%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 22.4%

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

   13. Applied rewrites 32.8%

       \[\leadsto \left(a \cdot \left(-b\right)\right) \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;a \cdot \left(x \cdot \left(-b\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(a, -x \cdot z, x\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{if}\;a \leq -2 \cdot 10^{-38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.06 \cdot 10^{+142}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* a (- (- z) b))))))
   (if (<= a -2e-38)
     t_1
     (if (<= a 1.06e+142) (* x (exp (* y (- (log z) t)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((a * (-z - b)));
	double tmp;
	if (a <= -2e-38) {
		tmp = t_1;
	} else if (a <= 1.06e+142) {
		tmp = x * exp((y * (log(z) - t)));
	} 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((a * (-z - b)))
    if (a <= (-2d-38)) then
        tmp = t_1
    else if (a <= 1.06d+142) then
        tmp = x * exp((y * (log(z) - t)))
    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((a * (-z - b)));
	double tmp;
	if (a <= -2e-38) {
		tmp = t_1;
	} else if (a <= 1.06e+142) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((a * (-z - b)))
	tmp = 0
	if a <= -2e-38:
		tmp = t_1
	elif a <= 1.06e+142:
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(a * Float64(Float64(-z) - b))))
	tmp = 0.0
	if (a <= -2e-38)
		tmp = t_1;
	elseif (a <= 1.06e+142)
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((a * (-z - b)));
	tmp = 0.0;
	if (a <= -2e-38)
		tmp = t_1;
	elseif (a <= 1.06e+142)
		tmp = x * exp((y * (log(z) - t)));
	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[(a * N[((-z) - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2e-38], t$95$1, If[LessEqual[a, 1.06e+142], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.9999999999999999e-38 or 1.06e142 < a

   1. Initial program 92.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.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. lower-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. lower-neg.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 88.9%

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

   if -1.9999999999999999e-38 < a < 1.06e142

   1. Initial program 98.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 89.6

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

   5. Applied rewrites 89.6%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\log z - t\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 74.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{if}\;a \leq -7.8 \cdot 10^{-47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 900000:\\ \;\;\;\;x \cdot e^{y \cdot \frac{1}{\frac{-1}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* a (- (- z) b))))))
   (if (<= a -7.8e-47)
     t_1
     (if (<= a 900000.0) (* x (exp (* y (/ 1.0 (/ -1.0 t))))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((a * (-z - b)));
	double tmp;
	if (a <= -7.8e-47) {
		tmp = t_1;
	} else if (a <= 900000.0) {
		tmp = x * exp((y * (1.0 / (-1.0 / t))));
	} 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((a * (-z - b)))
    if (a <= (-7.8d-47)) then
        tmp = t_1
    else if (a <= 900000.0d0) then
        tmp = x * exp((y * (1.0d0 / ((-1.0d0) / t))))
    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((a * (-z - b)));
	double tmp;
	if (a <= -7.8e-47) {
		tmp = t_1;
	} else if (a <= 900000.0) {
		tmp = x * Math.exp((y * (1.0 / (-1.0 / t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((a * (-z - b)))
	tmp = 0
	if a <= -7.8e-47:
		tmp = t_1
	elif a <= 900000.0:
		tmp = x * math.exp((y * (1.0 / (-1.0 / t))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(a * Float64(Float64(-z) - b))))
	tmp = 0.0
	if (a <= -7.8e-47)
		tmp = t_1;
	elseif (a <= 900000.0)
		tmp = Float64(x * exp(Float64(y * Float64(1.0 / Float64(-1.0 / t)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((a * (-z - b)));
	tmp = 0.0;
	if (a <= -7.8e-47)
		tmp = t_1;
	elseif (a <= 900000.0)
		tmp = x * exp((y * (1.0 / (-1.0 / t))));
	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[(a * N[((-z) - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -7.8e-47], t$95$1, If[LessEqual[a, 900000.0], N[(x * N[Exp[N[(y * N[(1.0 / N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.79999999999999956e-47 or 9e5 < a

   1. Initial program 93.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. lower-*.f64 N/A

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

      2. lower--.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. lower-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. lower-neg.f64 81.6

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 81.6%

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

   if -7.79999999999999956e-47 < a < 9e5

   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. lower-*.f64 N/A

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

      5. lower-neg.f64 81.1

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

   5. Applied rewrites 81.1%

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

   7. Applied rewrites 77.1%

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

   9. Applied rewrites 81.1%

      \[\leadsto x \cdot e^{y \cdot \frac{1}{\color{blue}{\frac{-1}{t}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 74.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{if}\;a \leq -7.8 \cdot 10^{-47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 900000:\\ \;\;\;\;x \cdot e^{-y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* a (- (- z) b))))))
   (if (<= a -7.8e-47) t_1 (if (<= a 900000.0) (* x (exp (- (* y t)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((a * (-z - b)));
	double tmp;
	if (a <= -7.8e-47) {
		tmp = t_1;
	} else if (a <= 900000.0) {
		tmp = x * exp(-(y * t));
	} 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((a * (-z - b)))
    if (a <= (-7.8d-47)) then
        tmp = t_1
    else if (a <= 900000.0d0) then
        tmp = x * exp(-(y * t))
    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((a * (-z - b)));
	double tmp;
	if (a <= -7.8e-47) {
		tmp = t_1;
	} else if (a <= 900000.0) {
		tmp = x * Math.exp(-(y * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((a * (-z - b)))
	tmp = 0
	if a <= -7.8e-47:
		tmp = t_1
	elif a <= 900000.0:
		tmp = x * math.exp(-(y * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(a * Float64(Float64(-z) - b))))
	tmp = 0.0
	if (a <= -7.8e-47)
		tmp = t_1;
	elseif (a <= 900000.0)
		tmp = Float64(x * exp(Float64(-Float64(y * t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((a * (-z - b)));
	tmp = 0.0;
	if (a <= -7.8e-47)
		tmp = t_1;
	elseif (a <= 900000.0)
		tmp = x * exp(-(y * t));
	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[(a * N[((-z) - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -7.8e-47], t$95$1, If[LessEqual[a, 900000.0], N[(x * N[Exp[(-N[(y * t), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.79999999999999956e-47 or 9e5 < a

   1. Initial program 93.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. lower-*.f64 N/A

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

      2. lower--.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. lower-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. lower-neg.f64 81.6

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 81.6%

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

   if -7.79999999999999956e-47 < a < 9e5

   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. lower-*.f64 N/A

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

      5. lower-neg.f64 81.1

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

   5. Applied rewrites 81.1%

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(-t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{-47}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{elif}\;a \leq 900000:\\ \;\;\;\;x \cdot e^{-y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{-y \cdot t}\\ \mathbf{if}\;t \leq -1060000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+39}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (- (* y t))))))
   (if (<= t -1060000000.0)
     t_1
     (if (<= t 8e+39) (* 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(-(y * t));
	double tmp;
	if (t <= -1060000000.0) {
		tmp = t_1;
	} else if (t <= 8e+39) {
		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(-(y * t))
    if (t <= (-1060000000.0d0)) then
        tmp = t_1
    else if (t <= 8d+39) 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(-(y * t));
	double tmp;
	if (t <= -1060000000.0) {
		tmp = t_1;
	} else if (t <= 8e+39) {
		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(-(y * t))
	tmp = 0
	if t <= -1060000000.0:
		tmp = t_1
	elif t <= 8e+39:
		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(-Float64(y * t))))
	tmp = 0.0
	if (t <= -1060000000.0)
		tmp = t_1;
	elseif (t <= 8e+39)
		tmp = Float64(x * exp(Float64(a * Float64(-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 * t));
	tmp = 0.0;
	if (t <= -1060000000.0)
		tmp = t_1;
	elseif (t <= 8e+39)
		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[(y * t), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1060000000.0], t$95$1, If[LessEqual[t, 8e+39], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.06e9 or 7.99999999999999952e39 < t

   1. Initial program 98.4%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      5. lower-neg.f64 84.6

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

   5. Applied rewrites 84.6%

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

   if -1.06e9 < t < 7.99999999999999952e39

   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 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. lower-neg.f64 N/A

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

      3. lower-*.f64 69.5

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

   5. Applied rewrites 69.5%

      \[\leadsto x \cdot e^{\color{blue}{-a \cdot b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1060000000:\\ \;\;\;\;x \cdot e^{-y \cdot t}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+39}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{a \cdot \left(-b\right)} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * exp((a * -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((a * -b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.3%

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

3. Taylor expanded in 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. lower-neg.f64 N/A

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

   3. lower-*.f64 58.2

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

5. Applied rewrites 58.2%

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

6. Final simplification 58.2%

   \[\leadsto x \cdot e^{a \cdot \left(-b\right)} \]
7. Add Preprocessing

Alternative 13: 17.6% accurate, 25.2× speedup

Math

\[\begin{array}{l} \\ -x \cdot \left(a \cdot b\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (- (* x (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return -(x * (a * 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 * (a * b))
end function

Java

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

Python

def code(x, y, z, t, a, b):
	return -(x * (a * b))

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := (-N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 96.3%

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

3. Taylor expanded in a around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. lower-fma.f64 N/A

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

5. Applied rewrites 68.9%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 26.2%

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

9. Taylor expanded in b around inf

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

11. Applied rewrites 16.3%

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

13. Applied rewrites 19.4%

    \[\leadsto \left(a \cdot \left(-b\right)\right) \cdot x \]

14. Final simplification 19.4%

    \[\leadsto -x \cdot \left(a \cdot b\right) \]
15. Add Preprocessing

Alternative 14: 17.1% accurate, 25.2× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(x \cdot \left(-b\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* a (* x (- b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return a * (x * -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 = a * (x * -b)
end function

Java

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

Python

def code(x, y, z, t, a, b):
	return a * (x * -b)

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(a * N[(x * (-b)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.3%

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

3. Taylor expanded in a around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. lower-fma.f64 N/A

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

5. Applied rewrites 68.9%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 26.2%

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

9. Taylor expanded in b around inf

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

11. Applied rewrites 16.3%

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

Reproduce

herbie shell --seed 2024219 
(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))))))