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

Percentage Accurate: 96.6% → 96.6%

Time: 10.6s

Alternatives: 8

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 4.3198617356903213e-191
  2. y = -2.2323163757528853e-79
  3. z = 2.3773204716526054e-203
  4. t = 2.857994010534309e-117
  5. a = 6.140127732529218e-217
  6. b = 1.7448144678286407e+277

• 42.0% 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: 96.6% 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: 96.6% 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 93.9%

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

Alternative 2: 32.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\\ \mathbf{if}\;t\_1 \leq 0 \lor \neg \left(t\_1 \leq 1\right):\\ \;\;\;\;x \cdot \left(\left(-t\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))
   (if (or (<= t_1 0.0) (not (<= t_1 1.0))) (* x (* (- t) y)) (* x 1.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
	double tmp;
	if ((t_1 <= 0.0) || !(t_1 <= 1.0)) {
		tmp = x * (-t * y);
	} 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 = exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
    if ((t_1 <= 0.0d0) .or. (.not. (t_1 <= 1.0d0))) then
        tmp = x * (-t * y)
    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 = Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
	double tmp;
	if ((t_1 <= 0.0) || !(t_1 <= 1.0)) {
		tmp = x * (-t * y);
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))
	tmp = 0
	if (t_1 <= 0.0) or not (t_1 <= 1.0):
		tmp = x * (-t * y)
	else:
		tmp = x * 1.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, If[Or[LessEqual[t$95$1, 0.0], N[Not[LessEqual[t$95$1, 1.0]], $MachinePrecision]], N[(x * N[((-t) * y), $MachinePrecision]), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))) < 0.0 or 1 < (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))))

   1. Initial program 95.8%

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

   3. Taylor expanded in 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 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 32.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 14.7%

       \[\leadsto x \cdot \mathsf{fma}\left(\log z - t, y, 1\right) \]

   12. Taylor expanded in t around inf

       \[\leadsto x \cdot \left(-1 \cdot \left(t \cdot y\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 16.4%

       \[\leadsto x \cdot \left(\left(-t\right) \cdot y\right) \]

   if 0.0 < (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))) < 1

   1. Initial program 85.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 84.4%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 83.8%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0 \lor \neg \left(e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 1\right):\\ \;\;\;\;x \cdot \left(\left(-t\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{-107} \lor \neg \left(y \leq 2.75 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\left(-z\right) - b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -6.6e-107) (not (<= y 2.75e-9)))
   (* x (pow (/ z (exp t)) y))
   (* x (exp (* (- (- z) b) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.6e-107) || !(y <= 2.75e-9)) {
		tmp = x * pow((z / exp(t)), y);
	} else {
		tmp = x * exp(((-z - 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 <= (-6.6d-107)) .or. (.not. (y <= 2.75d-9))) then
        tmp = x * ((z / exp(t)) ** y)
    else
        tmp = x * exp(((-z - 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 <= -6.6e-107) || !(y <= 2.75e-9)) {
		tmp = x * Math.pow((z / Math.exp(t)), y);
	} else {
		tmp = x * Math.exp(((-z - b) * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -6.6e-107) or not (y <= 2.75e-9):
		tmp = x * math.pow((z / math.exp(t)), y)
	else:
		tmp = x * math.exp(((-z - b) * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -6.6e-107) || !(y <= 2.75e-9))
		tmp = Float64(x * (Float64(z / exp(t)) ^ y));
	else
		tmp = Float64(x * exp(Float64(Float64(Float64(-z) - b) * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -6.6e-107) || ~((y <= 2.75e-9)))
		tmp = x * ((z / exp(t)) ^ y);
	else
		tmp = x * exp(((-z - b) * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -6.6e-107], N[Not[LessEqual[y, 2.75e-9]], $MachinePrecision]], N[(x * N[Power[N[(z / N[Exp[t], $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.60000000000000007e-107 or 2.7499999999999998e-9 < y

   1. Initial program 94.6%

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

   3. Taylor expanded in 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 88.1

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

   5. Applied rewrites 88.1%

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

   if -6.60000000000000007e-107 < y < 2.7499999999999998e-9

   1. Initial program 92.9%

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

   3. Taylor expanded in y around 0

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

      1. *-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 89.3

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 89.3%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{-107} \lor \neg \left(y \leq 2.75 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot {\left(\frac{z}{e^{t}}\right)}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\left(-z\right) - b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+25} \lor \neg \left(t \leq 5.1 \cdot 10^{+22}\right):\\ \;\;\;\;x \cdot e^{\left(-t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\left(-z\right) - b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -1.85e+25) (not (<= t 5.1e+22)))
   (* x (exp (* (- t) y)))
   (* x (exp (* (- (- z) b) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1.85e+25) || !(t <= 5.1e+22)) {
		tmp = x * exp((-t * y));
	} else {
		tmp = x * exp(((-z - 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 ((t <= (-1.85d+25)) .or. (.not. (t <= 5.1d+22))) then
        tmp = x * exp((-t * y))
    else
        tmp = x * exp(((-z - 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 ((t <= -1.85e+25) || !(t <= 5.1e+22)) {
		tmp = x * Math.exp((-t * y));
	} else {
		tmp = x * Math.exp(((-z - b) * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -1.85e+25) or not (t <= 5.1e+22):
		tmp = x * math.exp((-t * y))
	else:
		tmp = x * math.exp(((-z - b) * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -1.85e+25) || !(t <= 5.1e+22))
		tmp = Float64(x * exp(Float64(Float64(-t) * y)));
	else
		tmp = Float64(x * exp(Float64(Float64(Float64(-z) - b) * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -1.85e+25) || ~((t <= 5.1e+22)))
		tmp = x * exp((-t * y));
	else
		tmp = x * exp(((-z - b) * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -1.85e+25], N[Not[LessEqual[t, 5.1e+22]], $MachinePrecision]], N[(x * N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8499999999999999e25 or 5.1000000000000002e22 < t

   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 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 84.5

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

   5. Applied rewrites 84.5%

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

   if -1.8499999999999999e25 < t < 5.1000000000000002e22

   1. Initial program 93.2%

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

   3. Taylor expanded in y around 0

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

      1. *-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 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 75.4%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+25} \lor \neg \left(t \leq 5.1 \cdot 10^{+22}\right):\\ \;\;\;\;x \cdot e^{\left(-t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\left(-z\right) - b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+25} \lor \neg \left(t \leq 5.1 \cdot 10^{+22}\right):\\ \;\;\;\;x \cdot e^{\left(-t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -1.85e+25) (not (<= t 5.1e+22)))
   (* x (exp (* (- t) y)))
   (* x (exp (* (- b) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1.85e+25) || !(t <= 5.1e+22)) {
		tmp = x * exp((-t * 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 ((t <= (-1.85d+25)) .or. (.not. (t <= 5.1d+22))) then
        tmp = x * exp((-t * 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 ((t <= -1.85e+25) || !(t <= 5.1e+22)) {
		tmp = x * Math.exp((-t * y));
	} else {
		tmp = x * Math.exp((-b * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -1.85e+25) or not (t <= 5.1e+22):
		tmp = x * math.exp((-t * y))
	else:
		tmp = x * math.exp((-b * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -1.85e+25) || !(t <= 5.1e+22))
		tmp = Float64(x * exp(Float64(Float64(-t) * 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 ((t <= -1.85e+25) || ~((t <= 5.1e+22)))
		tmp = x * exp((-t * y));
	else
		tmp = x * exp((-b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -1.85e+25], N[Not[LessEqual[t, 5.1e+22]], $MachinePrecision]], N[(x * N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8499999999999999e25 or 5.1000000000000002e22 < t

   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 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 84.5

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

   5. Applied rewrites 84.5%

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

   if -1.8499999999999999e25 < t < 5.1000000000000002e22

   1. Initial program 93.2%

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

   3. Taylor expanded in y around 0

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

      1. *-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 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 66.6%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+25} \lor \neg \left(t \leq 5.1 \cdot 10^{+22}\right):\\ \;\;\;\;x \cdot e^{\left(-t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+73} \lor \neg \left(y \leq 145000\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.55e+73) (not (<= y 145000.0)))
   (* 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 <= -1.55e+73) || !(y <= 145000.0)) {
		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 <= (-1.55d+73)) .or. (.not. (y <= 145000.0d0))) 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 <= -1.55e+73) || !(y <= 145000.0)) {
		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 <= -1.55e+73) or not (y <= 145000.0):
		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 <= -1.55e+73) || !(y <= 145000.0))
		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 <= -1.55e+73) || ~((y <= 145000.0)))
		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, -1.55e+73], N[Not[LessEqual[y, 145000.0]], $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 < -1.55e73 or 145000 < y

   1. Initial program 93.4%

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

   3. Taylor expanded in 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 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 71.2%

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

   if -1.55e73 < y < 145000

   1. Initial program 94.2%

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

   3. Taylor expanded in y around 0

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

      1. *-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 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 74.2%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+73} \lor \neg \left(y \leq 145000\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{+34}:\\ \;\;\;\;x \cdot \left(\left(-t\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -6.4e+34) (* x (* (- t) y)) (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -6.4e+34) {
		tmp = x * (-t * y);
	} else {
		tmp = x * pow(z, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= (-6.4d+34)) then
        tmp = x * (-t * y)
    else
        tmp = x * (z ** y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -6.4e+34) {
		tmp = x * (-t * y);
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -6.4e+34:
		tmp = x * (-t * y)
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -6.4e+34)
		tmp = Float64(x * Float64(Float64(-t) * y));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -6.4e+34)
		tmp = x * (-t * y);
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -6.4e+34], N[(x * N[((-t) * y), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.3999999999999997e34

   1. Initial program 93.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{e^{y \cdot \left(\log z - t\right)}} \]
   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 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 10.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 25.2%

       \[\leadsto x \cdot \mathsf{fma}\left(\log z - t, y, 1\right) \]

   12. Taylor expanded in t around inf

       \[\leadsto x \cdot \left(-1 \cdot \left(t \cdot y\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 19.1%

       \[\leadsto x \cdot \left(\left(-t\right) \cdot y\right) \]

   if -6.3999999999999997e34 < 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 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 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 60.6%

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

Alternative 8: 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 93.9%

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

3. Taylor expanded in a around 0

   \[\leadsto x \cdot \color{blue}{e^{y \cdot \left(\log z - t\right)}} \]
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 71.2

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

5. Applied rewrites 71.2%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 48.7%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 18.3%

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

Reproduce

herbie shell --seed 2024318 
(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))))))