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

Percentage Accurate: 96.7% → 96.5%

Time: 11.7s

Alternatives: 12

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 4.35875512463875e+113
  2. y = -3.5449511120069664e-120
  3. z = 3.3086566762700965e-57
  4. t = 6.952099038051516e+267
  5. a = -1.2831205051962513e-111
  6. b = -9.731872020992759e+72

• 41.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+188}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \left(\log z - t\right) \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -2e+188)
   (* x (exp (* (- a) (+ z b))))
   (* x (exp (fma (- b) a (* (- (log z) t) y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2e+188) {
		tmp = x * exp((-a * (z + b)));
	} else {
		tmp = x * exp(fma(-b, a, ((log(z) - t) * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -2e+188)
		tmp = Float64(x * exp(Float64(Float64(-a) * Float64(z + b))));
	else
		tmp = Float64(x * exp(fma(Float64(-b), a, Float64(Float64(log(z) - t) * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -2e+188], N[(x * N[Exp[N[((-a) * N[(z + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-b) * a + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2e188

   1. Initial program 82.9%

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

   3. Taylor expanded in y around 0

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

      1. *-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 96.6

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

   5. Applied rewrites 96.6%

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

   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 96.6%

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

   if -2e188 < a

   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 z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 97.8

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

   5. Applied rewrites 97.8%

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

Alternative 2: 32.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+22} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+21}\right):\\ \;\;\;\;\left(-a\right) \cdot \left(b \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (or (<= t_1 -5e+22) (not (<= t_1 5e+21))) (* (- a) (* b x)) (* x 1.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if ((t_1 <= -5e+22) || !(t_1 <= 5e+21)) {
		tmp = -a * (b * x);
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if ((t_1 <= (-5d+22)) .or. (.not. (t_1 <= 5d+21))) then
        tmp = -a * (b * x)
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if ((t_1 <= -5e+22) || !(t_1 <= 5e+21)) {
		tmp = -a * (b * x);
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if (t_1 <= -5e+22) or not (t_1 <= 5e+21):
		tmp = -a * (b * x)
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if ((t_1 <= -5e+22) || !(t_1 <= 5e+21))
		tmp = Float64(Float64(-a) * Float64(b * x));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if ((t_1 <= -5e+22) || ~((t_1 <= 5e+21)))
		tmp = -a * (b * x);
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+22], N[Not[LessEqual[t$95$1, 5e+21]], $MachinePrecision]], N[((-a) * N[(b * x), $MachinePrecision]), $MachinePrecision], N[(x * 1.0), $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))) < -4.9999999999999996e22 or 5e21 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 96.5%

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

   3. Taylor expanded in 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 60.8%

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

   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 12.0%

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

   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 17.8%

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

   if -4.9999999999999996e22 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5e21

   1. Initial program 94.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. exp-diff N/A

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

      5. rem-exp-log N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 65.9

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 81.5%

      \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -5 \cdot 10^{+22} \lor \neg \left(y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+21}\right):\\ \;\;\;\;\left(-a\right) \cdot \left(b \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

Alternative 4: 32.2% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))) -5e+22)
   (* (- a) (* b x))
   (fma (- a) (fma z x (* b x)) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= -5e+22) {
		tmp = -a * (b * x);
	} else {
		tmp = fma(-a, fma(z, x, (b * x)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b))) <= -5e+22)
		tmp = Float64(Float64(-a) * Float64(b * x));
	else
		tmp = fma(Float64(-a), fma(z, x, Float64(b * x)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], -5e+22], N[((-a) * N[(b * x), $MachinePrecision]), $MachinePrecision], N[((-a) * N[(z * x + N[(b * x), $MachinePrecision]), $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))) < -4.9999999999999996e22

   1. Initial program 97.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. *-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 56.4%

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

   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 3.9%

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

   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 14.1%

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

   if -4.9999999999999996e22 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 95.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 65.7%

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

   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 44.4%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 45.0%

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

Alternative 5: 31.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\left(-a\right) \cdot \left(b \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(b \cdot x\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))) -5e+22)
   (* (- a) (* b x))
   (- x (* (* b x) a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= -5e+22) {
		tmp = -a * (b * x);
	} else {
		tmp = x - ((b * x) * a);
	}
	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 (((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))) <= (-5d+22)) then
        tmp = -a * (b * x)
    else
        tmp = x - ((b * x) * a)
    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 (((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))) <= -5e+22) {
		tmp = -a * (b * x);
	} else {
		tmp = x - ((b * x) * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))) <= -5e+22:
		tmp = -a * (b * x)
	else:
		tmp = x - ((b * x) * a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b))) <= -5e+22)
		tmp = Float64(Float64(-a) * Float64(b * x));
	else
		tmp = Float64(x - Float64(Float64(b * x) * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= -5e+22)
		tmp = -a * (b * x);
	else
		tmp = x - ((b * x) * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], -5e+22], N[((-a) * N[(b * x), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(b * x), $MachinePrecision] * a), $MachinePrecision]), $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))) < -4.9999999999999996e22

   1. Initial program 97.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. *-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 56.4%

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

   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 3.9%

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

   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 14.1%

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

   if -4.9999999999999996e22 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 95.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 65.7%

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

   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 44.4%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 45.0%

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

Alternative 6: 86.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.55 \cdot 10^{-57} \lor \neg \left(y \leq 1.25 \cdot 10^{+20}\right):\\ \;\;\;\;x \cdot e^{\left(\log z - t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot \left(z + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -4.55e-57) (not (<= y 1.25e+20)))
   (* x (exp (* (- (log z) t) y)))
   (* x (exp (* (- a) (+ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -4.55e-57) || !(y <= 1.25e+20)) {
		tmp = x * exp(((log(z) - t) * y));
	} else {
		tmp = x * exp((-a * (z + b)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-4.55d-57)) .or. (.not. (y <= 1.25d+20))) then
        tmp = x * exp(((log(z) - t) * y))
    else
        tmp = x * exp((-a * (z + b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -4.55e-57) || !(y <= 1.25e+20)) {
		tmp = x * Math.exp(((Math.log(z) - t) * y));
	} else {
		tmp = x * Math.exp((-a * (z + b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -4.55e-57) or not (y <= 1.25e+20):
		tmp = x * math.exp(((math.log(z) - t) * y))
	else:
		tmp = x * math.exp((-a * (z + b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -4.55e-57) || !(y <= 1.25e+20))
		tmp = Float64(x * exp(Float64(Float64(log(z) - t) * y)));
	else
		tmp = Float64(x * exp(Float64(Float64(-a) * Float64(z + b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -4.55e-57) || ~((y <= 1.25e+20)))
		tmp = x * exp(((log(z) - t) * y));
	else
		tmp = x * exp((-a * (z + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -4.55e-57], N[Not[LessEqual[y, 1.25e+20]], $MachinePrecision]], N[(x * N[Exp[N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-a) * N[(z + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.55000000000000017e-57 or 1.25e20 < y

   1. Initial program 96.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. exp-diff N/A

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

      5. rem-exp-log N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 85.2

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

   5. Applied rewrites 85.2%

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

   7. Applied rewrites 86.0%

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

   if -4.55000000000000017e-57 < y < 1.25e20

   1. Initial program 95.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   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 91.3

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

   5. Applied rewrites 91.3%

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

   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 91.3%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.55 \cdot 10^{-57} \lor \neg \left(y \leq 1.25 \cdot 10^{+20}\right):\\ \;\;\;\;x \cdot e^{\left(\log z - t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot \left(z + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{\left(-t\right) \cdot y}\\ \mathbf{if}\;t \leq -120:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-268}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+94}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* (- t) y)))))
   (if (<= t -120.0)
     t_1
     (if (<= t -1.3e-268)
       (* x (pow z y))
       (if (<= t 1.05e+94) (* x (exp (* (- a) (+ z b)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((-t * y));
	double tmp;
	if (t <= -120.0) {
		tmp = t_1;
	} else if (t <= -1.3e-268) {
		tmp = x * pow(z, y);
	} else if (t <= 1.05e+94) {
		tmp = x * exp((-a * (z + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * exp((-t * y))
    if (t <= (-120.0d0)) then
        tmp = t_1
    else if (t <= (-1.3d-268)) then
        tmp = x * (z ** y)
    else if (t <= 1.05d+94) then
        tmp = x * exp((-a * (z + b)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((-t * y));
	double tmp;
	if (t <= -120.0) {
		tmp = t_1;
	} else if (t <= -1.3e-268) {
		tmp = x * Math.pow(z, y);
	} else if (t <= 1.05e+94) {
		tmp = x * Math.exp((-a * (z + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((-t * y))
	tmp = 0
	if t <= -120.0:
		tmp = t_1
	elif t <= -1.3e-268:
		tmp = x * math.pow(z, y)
	elif t <= 1.05e+94:
		tmp = x * math.exp((-a * (z + b)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(Float64(-t) * y)))
	tmp = 0.0
	if (t <= -120.0)
		tmp = t_1;
	elseif (t <= -1.3e-268)
		tmp = Float64(x * (z ^ y));
	elseif (t <= 1.05e+94)
		tmp = Float64(x * exp(Float64(Float64(-a) * Float64(z + b))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((-t * y));
	tmp = 0.0;
	if (t <= -120.0)
		tmp = t_1;
	elseif (t <= -1.3e-268)
		tmp = x * (z ^ y);
	elseif (t <= 1.05e+94)
		tmp = x * exp((-a * (z + b)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -120.0], t$95$1, If[LessEqual[t, -1.3e-268], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.05e+94], N[(x * N[Exp[N[((-a) * N[(z + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -120 or 1.04999999999999995e94 < t

   1. Initial program 94.0%

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

   3. Taylor expanded in t around inf

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

      1. 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 77.1

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

   5. Applied rewrites 77.1%

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

   if -120 < t < -1.30000000000000001e-268

   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

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

      1. *-commutative N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. exp-diff N/A

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

      5. rem-exp-log N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 76.8%

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

   if -1.30000000000000001e-268 < t < 1.04999999999999995e94

   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 y around 0

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

      1. *-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 82.8

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

   5. Applied rewrites 82.8%

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

   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 82.8%

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

Alternative 8: 72.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{\left(-t\right) \cdot y}\\ \mathbf{if}\;t \leq -120:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-268}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;t \leq 1.66 \cdot 10^{+71}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* (- t) y)))))
   (if (<= t -120.0)
     t_1
     (if (<= t -1.1e-268)
       (* x (pow z y))
       (if (<= t 1.66e+71) (* x (exp (* (- b) a))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((-t * y));
	double tmp;
	if (t <= -120.0) {
		tmp = t_1;
	} else if (t <= -1.1e-268) {
		tmp = x * pow(z, y);
	} else if (t <= 1.66e+71) {
		tmp = x * exp((-b * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * exp((-t * y))
    if (t <= (-120.0d0)) then
        tmp = t_1
    else if (t <= (-1.1d-268)) then
        tmp = x * (z ** y)
    else if (t <= 1.66d+71) then
        tmp = x * exp((-b * a))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((-t * y));
	double tmp;
	if (t <= -120.0) {
		tmp = t_1;
	} else if (t <= -1.1e-268) {
		tmp = x * Math.pow(z, y);
	} else if (t <= 1.66e+71) {
		tmp = x * Math.exp((-b * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((-t * y))
	tmp = 0
	if t <= -120.0:
		tmp = t_1
	elif t <= -1.1e-268:
		tmp = x * math.pow(z, y)
	elif t <= 1.66e+71:
		tmp = x * math.exp((-b * a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(Float64(-t) * y)))
	tmp = 0.0
	if (t <= -120.0)
		tmp = t_1;
	elseif (t <= -1.1e-268)
		tmp = Float64(x * (z ^ y));
	elseif (t <= 1.66e+71)
		tmp = Float64(x * exp(Float64(Float64(-b) * a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((-t * y));
	tmp = 0.0;
	if (t <= -120.0)
		tmp = t_1;
	elseif (t <= -1.1e-268)
		tmp = x * (z ^ y);
	elseif (t <= 1.66e+71)
		tmp = x * exp((-b * a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -120.0], t$95$1, If[LessEqual[t, -1.1e-268], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.66e+71], N[(x * N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -120 or 1.65999999999999995e71 < t

   1. Initial program 93.3%

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

   3. Taylor expanded in t around inf

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

      1. 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 76.8

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

   5. Applied rewrites 76.8%

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

   if -120 < t < -1.10000000000000002e-268

   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

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

      1. *-commutative N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. exp-diff N/A

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

      5. rem-exp-log N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 76.8%

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

   if -1.10000000000000002e-268 < t < 1.65999999999999995e71

   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 z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 96.0

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 78.2%

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

Alternative 9: 73.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \lor \neg \left(y \leq 1950\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.4) (not (<= y 1950.0)))
   (* x (pow z y))
   (* x (exp (* (- b) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.4) || !(y <= 1950.0)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp((-b * a));
	}
	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 ((y <= (-2.4d0)) .or. (.not. (y <= 1950.0d0))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp((-b * a))
    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 ((y <= -2.4) || !(y <= 1950.0)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((-b * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -2.4) or not (y <= 1950.0):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((-b * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2.4) || !(y <= 1950.0))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(Float64(-b) * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -2.4) || ~((y <= 1950.0)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((-b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2.4], N[Not[LessEqual[y, 1950.0]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.39999999999999991 or 1950 < y

   1. Initial program 96.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. exp-diff N/A

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

      5. rem-exp-log N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 59.2%

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

   if -2.39999999999999991 < y < 1950

   1. Initial program 95.6%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 94.1

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 81.4%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \lor \neg \left(y \leq 1950\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -15.5 \lor \neg \left(y \leq 1850\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-z\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -15.5) (not (<= y 1850.0)))
   (* x (pow z y))
   (* x (exp (* (- z) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -15.5) || !(y <= 1850.0)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * exp((-z * a));
	}
	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 ((y <= (-15.5d0)) .or. (.not. (y <= 1850.0d0))) then
        tmp = x * (z ** y)
    else
        tmp = x * exp((-z * a))
    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 ((y <= -15.5) || !(y <= 1850.0)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((-z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -15.5) or not (y <= 1850.0):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((-z * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -15.5) || !(y <= 1850.0))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(Float64(-z) * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -15.5) || ~((y <= 1850.0)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((-z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -15.5], N[Not[LessEqual[y, 1850.0]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-z) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -15.5 or 1850 < y

   1. Initial program 96.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. exp-diff N/A

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

      5. rem-exp-log N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 59.2%

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

   if -15.5 < y < 1850

   1. Initial program 95.6%

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

   3. Taylor expanded in y around 0

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

      1. *-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 89.4

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

   5. Applied rewrites 89.4%

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

   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 89.4%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 56.7%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15.5 \lor \neg \left(y \leq 1850\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-z\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1100:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, a, \left(\left(z \cdot x\right) \cdot a\right) \cdot -0.5\right), z, x - \left(b \cdot x\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1100.0)
   (fma (fma (- x) a (* (* (* z x) a) -0.5)) z (- x (* (* b x) a)))
   (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1100.0) {
		tmp = fma(fma(-x, a, (((z * x) * a) * -0.5)), z, (x - ((b * x) * a)));
	} else {
		tmp = x * pow(z, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1100.0)
		tmp = fma(fma(Float64(-x), a, Float64(Float64(Float64(z * x) * a) * -0.5)), z, Float64(x - Float64(Float64(b * x) * a)));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1100.0], N[(N[((-x) * a + N[(N[(N[(z * x), $MachinePrecision] * a), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * z + N[(x - N[(N[(b * x), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1100

   1. Initial program 95.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. *-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 67.8%

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

   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 23.9%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 26.3%

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

   if -1100 < t

   1. Initial program 96.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. exp-diff N/A

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

      5. rem-exp-log N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 64.7

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 59.3%

      \[\leadsto x \cdot {z}^{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 19.2% accurate, 54.7× speedup

Math

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

FPCore

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

C

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

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 * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * 1.0), $MachinePrecision]

Derivation

1. Initial program 96.1%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. exp-prod N/A

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

   3. lower-pow.f64 N/A

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

   4. exp-diff N/A

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

   5. rem-exp-log N/A

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

   6. lower-/.f64 N/A

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

   7. lower-exp.f64 67.9

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

5. Applied rewrites 67.9%

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

6. Taylor expanded in y around 0

   \[\leadsto x \cdot 1 \]
7. Step-by-step derivation

8. Applied rewrites 20.5%

   \[\leadsto x \cdot 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024313 
(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))))))