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

Percentage Accurate: 96.4% → 96.4%

Time: 14.0s

Alternatives: 13

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 2.2962625187389004e+184
  2. y = -41714.03482347881
  3. z = 2.123778798078947e-110
  4. t = 1166475182.5174239
  5. a = 5.24087630483847e-20
  6. b = 1.9758053861028208e+306

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.4% 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: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Final simplification 98.0%

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

Alternative 2: 43.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y\\ t_2 := \mathsf{fma}\left(b \cdot a, -x, x\right)\\ \mathbf{if}\;t\_1 \leq -50:\\ \;\;\;\;\mathsf{log1p}\left(-z\right) \cdot \left(a \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+249}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, b, \left(\left(\mathsf{fma}\left(-0.3333333333333333, x, \frac{\mathsf{fma}\left(0.5, x, \frac{x}{z}\right)}{-z}\right) \cdot z\right) \cdot z\right) \cdot z\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* (- (log (- 1.0 z)) b) a) (* (- (log z) t) y)))
        (t_2 (fma (* b a) (- x) x)))
   (if (<= t_1 -50.0)
     (* (log1p (- z)) (* a x))
     (if (<= t_1 1e-7)
       t_2
       (if (<= t_1 1e+249)
         (fma
          (fma
           (- x)
           b
           (*
            (*
             (* (fma -0.3333333333333333 x (/ (fma 0.5 x (/ x z)) (- z))) z)
             z)
            z))
          a
          x)
         t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((log((1.0 - z)) - b) * a) + ((log(z) - t) * y);
	double t_2 = fma((b * a), -x, x);
	double tmp;
	if (t_1 <= -50.0) {
		tmp = log1p(-z) * (a * x);
	} else if (t_1 <= 1e-7) {
		tmp = t_2;
	} else if (t_1 <= 1e+249) {
		tmp = fma(fma(-x, b, (((fma(-0.3333333333333333, x, (fma(0.5, x, (x / z)) / -z)) * z) * z) * z)), a, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(log(Float64(1.0 - z)) - b) * a) + Float64(Float64(log(z) - t) * y))
	t_2 = fma(Float64(b * a), Float64(-x), x)
	tmp = 0.0
	if (t_1 <= -50.0)
		tmp = Float64(log1p(Float64(-z)) * Float64(a * x));
	elseif (t_1 <= 1e-7)
		tmp = t_2;
	elseif (t_1 <= 1e+249)
		tmp = fma(fma(Float64(-x), b, Float64(Float64(Float64(fma(-0.3333333333333333, x, Float64(fma(0.5, x, Float64(x / z)) / Float64(-z))) * z) * z) * z)), a, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b * a), $MachinePrecision] * (-x) + x), $MachinePrecision]}, If[LessEqual[t$95$1, -50.0], N[(N[Log[1 + (-z)], $MachinePrecision] * N[(a * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], t$95$2, If[LessEqual[t$95$1, 1e+249], N[(N[((-x) * b + N[(N[(N[(N[(-0.3333333333333333 * x + N[(N[(0.5 * x + N[(x / z), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], t$95$2]]]]]

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))) < -50

   1. Initial program 97.9%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 59.3%

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

   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 3.8%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 4.4%

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

   12. Taylor expanded in a around inf

       \[\leadsto a \cdot \left(x \cdot \log \left(1 - z\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 26.8%

       \[\leadsto \left(x \cdot a\right) \cdot \mathsf{log1p}\left(-z\right) \]

   if -50 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.9999999999999995e-8 or 9.9999999999999992e248 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 98.9%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 60.1%

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

   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 67.0%

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

   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 61.1%

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

   12. Taylor expanded in z around 0

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

   14. Applied rewrites 73.1%

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

   if 9.9999999999999995e-8 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.9999999999999992e248

   1. Initial program 97.0%

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

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 71.4%

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

   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 15.2%

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

   9. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot \left(b \cdot x\right) + z \cdot \left(-1 \cdot x + z \cdot \left(\frac{-1}{2} \cdot x + \frac{-1}{3} \cdot \left(x \cdot z\right)\right)\right), a, x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 15.2%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-x, b, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot x, -0.3333333333333333, -0.5 \cdot x\right), z, -x\right) \cdot z\right), a, x\right) \]

   12. Taylor expanded in z around -inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-x, b, \left({z}^{2} \cdot \left(-1 \cdot \frac{\frac{1}{2} \cdot x + \frac{x}{z}}{z} + \frac{-1}{3} \cdot x\right)\right) \cdot z\right), a, x\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 27.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-x, b, \left(\left(\mathsf{fma}\left(-0.3333333333333333, x, \frac{\mathsf{fma}\left(0.5, x, \frac{x}{z}\right)}{-z}\right) \cdot z\right) \cdot z\right) \cdot z\right), a, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq -50:\\ \;\;\;\;\mathsf{log1p}\left(-z\right) \cdot \left(a \cdot x\right)\\ \mathbf{elif}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, -x, x\right)\\ \mathbf{elif}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq 10^{+249}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, b, \left(\left(\mathsf{fma}\left(-0.3333333333333333, x, \frac{\mathsf{fma}\left(0.5, x, \frac{x}{z}\right)}{-z}\right) \cdot z\right) \cdot z\right) \cdot z\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, -x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 36.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y\\ t_2 := \mathsf{fma}\left(b \cdot a, -x, x\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{a} \cdot a\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+252}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, b, \left(\left(\mathsf{fma}\left(-0.3333333333333333, x, \frac{\mathsf{fma}\left(0.5, x, \frac{x}{z}\right)}{-z}\right) \cdot z\right) \cdot z\right) \cdot z\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* (- (log (- 1.0 z)) b) a) (* (- (log z) t) y)))
        (t_2 (fma (* b a) (- x) x)))
   (if (<= t_1 -1e-8)
     (* (/ x a) a)
     (if (<= t_1 1e-7)
       t_2
       (if (<= t_1 1e+252)
         (fma
          (fma
           (- x)
           b
           (*
            (*
             (* (fma -0.3333333333333333 x (/ (fma 0.5 x (/ x z)) (- z))) z)
             z)
            z))
          a
          x)
         t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((log((1.0 - z)) - b) * a) + ((log(z) - t) * y);
	double t_2 = fma((b * a), -x, x);
	double tmp;
	if (t_1 <= -1e-8) {
		tmp = (x / a) * a;
	} else if (t_1 <= 1e-7) {
		tmp = t_2;
	} else if (t_1 <= 1e+252) {
		tmp = fma(fma(-x, b, (((fma(-0.3333333333333333, x, (fma(0.5, x, (x / z)) / -z)) * z) * z) * z)), a, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(log(Float64(1.0 - z)) - b) * a) + Float64(Float64(log(z) - t) * y))
	t_2 = fma(Float64(b * a), Float64(-x), x)
	tmp = 0.0
	if (t_1 <= -1e-8)
		tmp = Float64(Float64(x / a) * a);
	elseif (t_1 <= 1e-7)
		tmp = t_2;
	elseif (t_1 <= 1e+252)
		tmp = fma(fma(Float64(-x), b, Float64(Float64(Float64(fma(-0.3333333333333333, x, Float64(fma(0.5, x, Float64(x / z)) / Float64(-z))) * z) * z) * z)), a, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b * a), $MachinePrecision] * (-x) + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-8], N[(N[(x / a), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], t$95$2, If[LessEqual[t$95$1, 1e+252], N[(N[((-x) * b + N[(N[(N[(N[(-0.3333333333333333 * x + N[(N[(0.5 * x + N[(x / z), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], t$95$2]]]]]

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))) < -1e-8

   1. Initial program 98.0%

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

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 58.7%

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

   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 4.4%

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

   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 5.3%

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

   12. Taylor expanded in a around 0

       \[\leadsto \frac{x}{a} \cdot a \]
   13. Step-by-step derivation

   14. Applied rewrites 18.5%

       \[\leadsto \frac{x}{a} \cdot a \]

   if -1e-8 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.9999999999999995e-8 or 1.0000000000000001e252 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 98.8%

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

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 60.9%

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

   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 68.5%

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

   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 62.4%

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

   12. Taylor expanded in z around 0

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

   14. Applied rewrites 74.8%

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

   if 9.9999999999999995e-8 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.0000000000000001e252

   1. Initial program 97.1%

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

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 70.8%

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

   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 14.8%

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

   9. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot \left(b \cdot x\right) + z \cdot \left(-1 \cdot x + z \cdot \left(\frac{-1}{2} \cdot x + \frac{-1}{3} \cdot \left(x \cdot z\right)\right)\right), a, x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 14.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-x, b, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot x, -0.3333333333333333, -0.5 \cdot x\right), z, -x\right) \cdot z\right), a, x\right) \]

   12. Taylor expanded in z around -inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-x, b, \left({z}^{2} \cdot \left(-1 \cdot \frac{\frac{1}{2} \cdot x + \frac{x}{z}}{z} + \frac{-1}{3} \cdot x\right)\right) \cdot z\right), a, x\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 26.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-x, b, \left(\left(\mathsf{fma}\left(-0.3333333333333333, x, \frac{\mathsf{fma}\left(0.5, x, \frac{x}{z}\right)}{-z}\right) \cdot z\right) \cdot z\right) \cdot z\right), a, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{a} \cdot a\\ \mathbf{elif}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, -x, x\right)\\ \mathbf{elif}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq 10^{+252}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, b, \left(\left(\mathsf{fma}\left(-0.3333333333333333, x, \frac{\mathsf{fma}\left(0.5, x, \frac{x}{z}\right)}{-z}\right) \cdot z\right) \cdot z\right) \cdot z\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, -x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 33.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+26}:\\ \;\;\;\;\left(\left(-x\right) \cdot b\right) \cdot a\\ \mathbf{elif}\;t\_1 \leq 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\right) \cdot z, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot a\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* (- (log (- 1.0 z)) b) a) (* (- (log z) t) y))))
   (if (<= t_1 -5e+26)
     (* (* (- x) b) a)
     (if (<= t_1 1e+88) (fma (* (- x) z) a x) (* (* (- b) a) x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((log((1.0 - z)) - b) * a) + ((log(z) - t) * y);
	double tmp;
	if (t_1 <= -5e+26) {
		tmp = (-x * b) * a;
	} else if (t_1 <= 1e+88) {
		tmp = fma((-x * z), a, x);
	} else {
		tmp = (-b * a) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(log(Float64(1.0 - z)) - b) * a) + Float64(Float64(log(z) - t) * y))
	tmp = 0.0
	if (t_1 <= -5e+26)
		tmp = Float64(Float64(Float64(-x) * b) * a);
	elseif (t_1 <= 1e+88)
		tmp = fma(Float64(Float64(-x) * z), a, x);
	else
		tmp = Float64(Float64(Float64(-b) * a) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+26], N[(N[((-x) * b), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[t$95$1, 1e+88], N[(N[((-x) * z), $MachinePrecision] * a + x), $MachinePrecision], N[(N[((-b) * a), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -5.0000000000000001e26

   1. Initial program 97.8%

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

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 62.0%

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

   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 3.9%

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

   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 17.9%

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

   if -5.0000000000000001e26 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.99999999999999959e87

   1. Initial program 99.9%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 59.2%

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

   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 62.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 62.5%

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

   12. Taylor expanded in z around 0

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

   14. Applied rewrites 63.8%

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

   if 9.99999999999999959e87 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 96.6%

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

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 66.3%

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

   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 28.6%

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

   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 27.0%

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

   13. Applied rewrites 30.2%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq -5 \cdot 10^{+26}:\\ \;\;\;\;\left(\left(-x\right) \cdot b\right) \cdot a\\ \mathbf{elif}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\right) \cdot z, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot a\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 32.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{a} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, -x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* (- (log (- 1.0 z)) b) a) (* (- (log z) t) y)) -1e-8)
   (* (/ x a) a)
   (fma (* b a) (- x) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((log((1.0 - z)) - b) * a) + ((log(z) - t) * y)) <= -1e-8) {
		tmp = (x / a) * a;
	} else {
		tmp = fma((b * a), -x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(log(Float64(1.0 - z)) - b) * a) + Float64(Float64(log(z) - t) * y)) <= -1e-8)
		tmp = Float64(Float64(x / a) * a);
	else
		tmp = fma(Float64(b * a), Float64(-x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], -1e-8], N[(N[(x / a), $MachinePrecision] * a), $MachinePrecision], N[(N[(b * a), $MachinePrecision] * (-x) + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e-8

   1. Initial program 98.0%

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

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 58.7%

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

   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 4.4%

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

   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 5.3%

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

   12. Taylor expanded in a around 0

       \[\leadsto \frac{x}{a} \cdot a \]
   13. Step-by-step derivation

   14. Applied rewrites 18.5%

       \[\leadsto \frac{x}{a} \cdot a \]

   if -1e-8 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 98.1%

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

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 65.1%

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

   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 45.7%

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

   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 45.3%

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

   12. Taylor expanded in z around 0

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

   14. Applied rewrites 48.2%

       \[\leadsto \mathsf{fma}\left(b \cdot a, -x, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{a} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, -x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq -5 \cdot 10^{+16}:\\ \;\;\;\;\left(\left(-x\right) \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, -x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* (- (log (- 1.0 z)) b) a) (* (- (log z) t) y)) -5e+16)
   (* (* (- x) b) a)
   (fma (* b a) (- x) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((log((1.0 - z)) - b) * a) + ((log(z) - t) * y)) <= -5e+16) {
		tmp = (-x * b) * a;
	} else {
		tmp = fma((b * a), -x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(log(Float64(1.0 - z)) - b) * a) + Float64(Float64(log(z) - t) * y)) <= -5e+16)
		tmp = Float64(Float64(Float64(-x) * b) * a);
	else
		tmp = fma(Float64(b * a), Float64(-x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], -5e+16], N[(N[((-x) * b), $MachinePrecision] * a), $MachinePrecision], N[(N[(b * a), $MachinePrecision] * (-x) + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -5e16

   1. Initial program 97.9%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 61.8%

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

   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 3.9%

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

   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 17.6%

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

   if -5e16 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 98.1%

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

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 63.2%

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

   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 44.7%

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

   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 44.3%

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

   12. Taylor expanded in z around 0

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

   14. Applied rewrites 47.1%

       \[\leadsto \mathsf{fma}\left(b \cdot a, -x, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq -5 \cdot 10^{+16}:\\ \;\;\;\;\left(\left(-x\right) \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, -x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 19.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq 5 \cdot 10^{+207}:\\ \;\;\;\;\left(\left(-x\right) \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot a\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* (- (log (- 1.0 z)) b) a) (* (- (log z) t) y)) 5e+207)
   (* (* (- x) b) a)
   (* (* (- b) a) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((log((1.0 - z)) - b) * a) + ((log(z) - t) * y)) <= 5e+207) {
		tmp = (-x * b) * a;
	} else {
		tmp = (-b * a) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((((log((1.0d0 - z)) - b) * a) + ((log(z) - t) * y)) <= 5d+207) then
        tmp = (-x * b) * a
    else
        tmp = (-b * a) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((Math.log((1.0 - z)) - b) * a) + ((Math.log(z) - t) * y)) <= 5e+207) {
		tmp = (-x * b) * a;
	} else {
		tmp = (-b * a) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (((math.log((1.0 - z)) - b) * a) + ((math.log(z) - t) * y)) <= 5e+207:
		tmp = (-x * b) * a
	else:
		tmp = (-b * a) * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(log(Float64(1.0 - z)) - b) * a) + Float64(Float64(log(z) - t) * y)) <= 5e+207)
		tmp = Float64(Float64(Float64(-x) * b) * a);
	else
		tmp = Float64(Float64(Float64(-b) * a) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((((log((1.0 - z)) - b) * a) + ((log(z) - t) * y)) <= 5e+207)
		tmp = (-x * b) * a;
	else
		tmp = (-b * a) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], 5e+207], N[(N[((-x) * b), $MachinePrecision] * a), $MachinePrecision], N[(N[((-b) * a), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.9999999999999999e207

   1. Initial program 98.0%

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

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 62.3%

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

   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 29.0%

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

   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 12.8%

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

   if 4.9999999999999999e207 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 98.2%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. exp-diff N/A

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

      10. rem-exp-log N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 63.9%

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

   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 33.2%

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

   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 30.9%

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

   13. Applied rewrites 39.4%

       \[\leadsto \left(\left(-a\right) \cdot b\right) \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y \leq 5 \cdot 10^{+207}:\\ \;\;\;\;\left(\left(-x\right) \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot a\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{-123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-15}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- (log z) t) y)) x)))
   (if (<= y -2.9e-123)
     t_1
     (if (<= y 5.1e-15) (* (exp (* (- (- z) b) a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp(((log(z) - t) * y)) * x;
	double tmp;
	if (y <= -2.9e-123) {
		tmp = t_1;
	} else if (y <= 5.1e-15) {
		tmp = exp(((-z - b) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp(((log(z) - t) * y)) * x
    if (y <= (-2.9d-123)) then
        tmp = t_1
    else if (y <= 5.1d-15) then
        tmp = exp(((-z - b) * a)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp(((Math.log(z) - t) * y)) * x;
	double tmp;
	if (y <= -2.9e-123) {
		tmp = t_1;
	} else if (y <= 5.1e-15) {
		tmp = Math.exp(((-z - b) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp(((math.log(z) - t) * y)) * x
	tmp = 0
	if y <= -2.9e-123:
		tmp = t_1
	elif y <= 5.1e-15:
		tmp = math.exp(((-z - b) * a)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(log(z) - t) * y)) * x)
	tmp = 0.0
	if (y <= -2.9e-123)
		tmp = t_1;
	elseif (y <= 5.1e-15)
		tmp = Float64(exp(Float64(Float64(Float64(-z) - b) * a)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp(((log(z) - t) * y)) * x;
	tmp = 0.0;
	if (y <= -2.9e-123)
		tmp = t_1;
	elseif (y <= 5.1e-15)
		tmp = exp(((-z - b) * a)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -2.9e-123], t$95$1, If[LessEqual[y, 5.1e-15], N[(N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.90000000000000004e-123 or 5.1e-15 < y

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 86.2

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

   5. Applied rewrites 86.2%

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

   if -2.90000000000000004e-123 < y < 5.1e-15

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f64 N/A

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

      6. lower-neg.f64 88.8

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 88.8%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{-123}:\\ \;\;\;\;e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-15}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log z - t\right) \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\log z \cdot y} \cdot x\\ \mathbf{if}\;y \leq -42000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.4:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (log z) y)) x)))
   (if (<= y -42000.0) t_1 (if (<= y 2.4) (* (exp (* (- (- z) b) a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((log(z) * y)) * x;
	double tmp;
	if (y <= -42000.0) {
		tmp = t_1;
	} else if (y <= 2.4) {
		tmp = exp(((-z - b) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp((log(z) * y)) * x
    if (y <= (-42000.0d0)) then
        tmp = t_1
    else if (y <= 2.4d0) then
        tmp = exp(((-z - b) * a)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp((Math.log(z) * y)) * x;
	double tmp;
	if (y <= -42000.0) {
		tmp = t_1;
	} else if (y <= 2.4) {
		tmp = Math.exp(((-z - b) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((math.log(z) * y)) * x
	tmp = 0
	if y <= -42000.0:
		tmp = t_1
	elif y <= 2.4:
		tmp = math.exp(((-z - b) * a)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(log(z) * y)) * x)
	tmp = 0.0
	if (y <= -42000.0)
		tmp = t_1;
	elseif (y <= 2.4)
		tmp = Float64(exp(Float64(Float64(Float64(-z) - b) * a)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((log(z) * y)) * x;
	tmp = 0.0;
	if (y <= -42000.0)
		tmp = t_1;
	elseif (y <= 2.4)
		tmp = exp(((-z - b) * a)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -42000.0], t$95$1, If[LessEqual[y, 2.4], N[(N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -42000 or 2.39999999999999991 < y

   1. Initial program 96.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 88.5

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

   5. Applied rewrites 88.5%

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

   6. Taylor expanded in t around 0

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\log z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.6%

      \[\leadsto x \cdot e^{\log z \cdot \color{blue}{y}} \]

   if -42000 < y < 2.39999999999999991

   1. Initial program 99.2%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f64 N/A

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

      6. lower-neg.f64 85.5

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 85.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -42000:\\ \;\;\;\;e^{\log z \cdot y} \cdot x\\ \mathbf{elif}\;y \leq 2.4:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\log z \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{if}\;t \leq -1.12 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-47}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- t) y)) x)))
   (if (<= t -1.12e+24)
     t_1
     (if (<= t 3.1e-47) (* (exp (* (- (- z) b) a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((-t * y)) * x;
	double tmp;
	if (t <= -1.12e+24) {
		tmp = t_1;
	} else if (t <= 3.1e-47) {
		tmp = exp(((-z - b) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp((-t * y)) * x
    if (t <= (-1.12d+24)) then
        tmp = t_1
    else if (t <= 3.1d-47) then
        tmp = exp(((-z - b) * a)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp((-t * y)) * x;
	double tmp;
	if (t <= -1.12e+24) {
		tmp = t_1;
	} else if (t <= 3.1e-47) {
		tmp = Math.exp(((-z - b) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((-t * y)) * x
	tmp = 0
	if t <= -1.12e+24:
		tmp = t_1
	elif t <= 3.1e-47:
		tmp = math.exp(((-z - b) * a)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(-t) * y)) * x)
	tmp = 0.0
	if (t <= -1.12e+24)
		tmp = t_1;
	elseif (t <= 3.1e-47)
		tmp = Float64(exp(Float64(Float64(Float64(-z) - b) * a)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((-t * y)) * x;
	tmp = 0.0;
	if (t <= -1.12e+24)
		tmp = t_1;
	elseif (t <= 3.1e-47)
		tmp = exp(((-z - b) * a)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -1.12e+24], t$95$1, If[LessEqual[t, 3.1e-47], N[(N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.12e24 or 3.0999999999999998e-47 < t

   1. Initial program 98.5%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 83.2

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

   5. Applied rewrites 83.2%

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

   if -1.12e24 < t < 3.0999999999999998e-47

   1. Initial program 97.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f64 N/A

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

      6. lower-neg.f64 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 74.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{+24}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-47}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{if}\;t \leq -1.12 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-47}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- t) y)) x)))
   (if (<= t -1.12e+24) t_1 (if (<= t 2.9e-47) (* (exp (* (- b) a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((-t * y)) * x;
	double tmp;
	if (t <= -1.12e+24) {
		tmp = t_1;
	} else if (t <= 2.9e-47) {
		tmp = exp((-b * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp((-t * y)) * x
    if (t <= (-1.12d+24)) then
        tmp = t_1
    else if (t <= 2.9d-47) then
        tmp = exp((-b * a)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp((-t * y)) * x;
	double tmp;
	if (t <= -1.12e+24) {
		tmp = t_1;
	} else if (t <= 2.9e-47) {
		tmp = Math.exp((-b * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((-t * y)) * x
	tmp = 0
	if t <= -1.12e+24:
		tmp = t_1
	elif t <= 2.9e-47:
		tmp = math.exp((-b * a)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(-t) * y)) * x)
	tmp = 0.0
	if (t <= -1.12e+24)
		tmp = t_1;
	elseif (t <= 2.9e-47)
		tmp = Float64(exp(Float64(Float64(-b) * a)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((-t * y)) * x;
	tmp = 0.0;
	if (t <= -1.12e+24)
		tmp = t_1;
	elseif (t <= 2.9e-47)
		tmp = exp((-b * a)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -1.12e+24], t$95$1, If[LessEqual[t, 2.9e-47], N[(N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.12e24 or 2.9e-47 < t

   1. Initial program 98.5%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 83.2

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

   5. Applied rewrites 83.2%

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

   if -1.12e24 < t < 2.9e-47

   1. Initial program 97.5%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 69.8

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

   5. Applied rewrites 69.8%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{+24}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-47}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp((-b * a)) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 61.3

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

5. Applied rewrites 61.3%

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

6. Final simplification 61.3%

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

Alternative 13: 18.0% accurate, 25.2× speedup

Math

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

FPCore

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

C

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

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 * b) * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. exp-diff N/A

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

   10. rem-exp-log N/A

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

   11. lower-/.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. lower-fma.f64 N/A

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

5. Applied rewrites 62.7%

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

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 29.9%

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

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 16.7%

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

12. Final simplification 16.7%

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

Reproduce

herbie shell --seed 2024255 
(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))))))