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

Percentage Accurate: 96.7% → 96.7%

Time: 11.0s

Alternatives: 8

Speedup: 1.0×

• 41.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Final simplification 96.2%

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

Alternative 2: 32.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* (* t y) x) 1.0)))
   (if (<= (- (* (- (log (- 1.0 z)) b) a) (* (- t (log z)) y)) -1e+21)
     (/ (- (* (* 1.0 x) (* 1.0 x)) (* t_1 t_1)) (+ t_1 (* 1.0 x)))
     (fma 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 = ((t * y) * x) * 1.0;
	double tmp;
	if ((((log((1.0 - z)) - b) * a) - ((t - log(z)) * y)) <= -1e+21) {
		tmp = (((1.0 * x) * (1.0 * x)) - (t_1 * t_1)) / (t_1 + (1.0 * x));
	} else {
		tmp = fma(1.0, x, (((-t * y) * x) * 1.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(t * y) * x) * 1.0)
	tmp = 0.0
	if (Float64(Float64(Float64(log(Float64(1.0 - z)) - b) * a) - Float64(Float64(t - log(z)) * y)) <= -1e+21)
		tmp = Float64(Float64(Float64(Float64(1.0 * x) * Float64(1.0 * x)) - Float64(t_1 * t_1)) / Float64(t_1 + Float64(1.0 * x)));
	else
		tmp = fma(1.0, x, Float64(Float64(Float64(Float64(-t) * y) * x) * 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(t * y), $MachinePrecision] * x), $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * a), $MachinePrecision] - N[(N[(t - N[Log[z], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], -1e+21], N[(N[(N[(N[(1.0 * x), $MachinePrecision] * N[(1.0 * x), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(1.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * x + N[(N[(N[((-t) * y), $MachinePrecision] * x), $MachinePrecision] * 1.0), $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))) < -1e21

   1. Initial program 99.1%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

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

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 31.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 3.0%

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

   13. Applied rewrites 13.1%

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

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

   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 t around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

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

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 43.9%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 38.9%

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

   13. Applied rewrites 40.4%

       \[\leadsto \mathsf{fma}\left(1, \color{blue}{x}, 1 \cdot \left(\left(-x\right) \cdot \left(t \cdot y\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.4%

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

Alternative 3: 84.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{if}\;y \leq -2.75 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+71}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- (log z) t) y)) x)))
   (if (<= y -2.75e+51)
     t_1
     (if (<= y 6.5e+71) (* (exp (* (- (- z) b) a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp(((log(z) - t) * y)) * x;
	double tmp;
	if (y <= -2.75e+51) {
		tmp = t_1;
	} else if (y <= 6.5e+71) {
		tmp = exp(((-z - b) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp(((log(z) - t) * y)) * x
    if (y <= (-2.75d+51)) then
        tmp = t_1
    else if (y <= 6.5d+71) then
        tmp = exp(((-z - b) * a)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp(((Math.log(z) - t) * y)) * x;
	double tmp;
	if (y <= -2.75e+51) {
		tmp = t_1;
	} else if (y <= 6.5e+71) {
		tmp = Math.exp(((-z - b) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp(((math.log(z) - t) * y)) * x
	tmp = 0
	if y <= -2.75e+51:
		tmp = t_1
	elif y <= 6.5e+71:
		tmp = math.exp(((-z - b) * a)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(log(z) - t) * y)) * x)
	tmp = 0.0
	if (y <= -2.75e+51)
		tmp = t_1;
	elseif (y <= 6.5e+71)
		tmp = Float64(exp(Float64(Float64(Float64(-z) - b) * a)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp(((log(z) - t) * y)) * x;
	tmp = 0.0;
	if (y <= -2.75e+51)
		tmp = t_1;
	elseif (y <= 6.5e+71)
		tmp = exp(((-z - b) * a)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -2.75e+51], t$95$1, If[LessEqual[y, 6.5e+71], N[(N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.75e51 or 6.49999999999999954e71 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if -2.75e51 < y < 6.49999999999999954e71

   1. Initial program 93.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f64 N/A

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

      6. lower-neg.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 84.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.75 \cdot 10^{+51}:\\ \;\;\;\;e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+71}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log z - t\right) \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{+56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+149}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- t) y)) x)))
   (if (<= t -1.3e+56)
     t_1
     (if (<= t 2.7e+149) (* (exp (* (- (- z) b) a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((-t * y)) * x;
	double tmp;
	if (t <= -1.3e+56) {
		tmp = t_1;
	} else if (t <= 2.7e+149) {
		tmp = exp(((-z - b) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp((-t * y)) * x
    if (t <= (-1.3d+56)) then
        tmp = t_1
    else if (t <= 2.7d+149) then
        tmp = exp(((-z - b) * a)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp((-t * y)) * x;
	double tmp;
	if (t <= -1.3e+56) {
		tmp = t_1;
	} else if (t <= 2.7e+149) {
		tmp = Math.exp(((-z - b) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((-t * y)) * x
	tmp = 0
	if t <= -1.3e+56:
		tmp = t_1
	elif t <= 2.7e+149:
		tmp = math.exp(((-z - b) * a)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(-t) * y)) * x)
	tmp = 0.0
	if (t <= -1.3e+56)
		tmp = t_1;
	elseif (t <= 2.7e+149)
		tmp = Float64(exp(Float64(Float64(Float64(-z) - b) * a)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((-t * y)) * x;
	tmp = 0.0;
	if (t <= -1.3e+56)
		tmp = t_1;
	elseif (t <= 2.7e+149)
		tmp = exp(((-z - b) * a)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -1.3e+56], t$95$1, If[LessEqual[t, 2.7e+149], N[(N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.30000000000000005e56 or 2.7000000000000001e149 < t

   1. Initial program 98.8%

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

   3. Taylor expanded in 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 87.3

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

   5. Applied rewrites 87.3%

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

   if -1.30000000000000005e56 < t < 2.7000000000000001e149

   1. Initial program 94.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f64 N/A

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

      6. lower-neg.f64 72.7

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

   5. Applied rewrites 72.7%

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

   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 72.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{+56}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+149}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{if}\;t \leq -6 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+149}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- t) y)) x)))
   (if (<= t -6e+59) t_1 (if (<= t 2.7e+149) (* (exp (* (- b) a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((-t * y)) * x;
	double tmp;
	if (t <= -6e+59) {
		tmp = t_1;
	} else if (t <= 2.7e+149) {
		tmp = exp((-b * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp((-t * y)) * x
    if (t <= (-6d+59)) then
        tmp = t_1
    else if (t <= 2.7d+149) then
        tmp = exp((-b * a)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp((-t * y)) * x;
	double tmp;
	if (t <= -6e+59) {
		tmp = t_1;
	} else if (t <= 2.7e+149) {
		tmp = Math.exp((-b * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((-t * y)) * x
	tmp = 0
	if t <= -6e+59:
		tmp = t_1
	elif t <= 2.7e+149:
		tmp = math.exp((-b * a)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(-t) * y)) * x)
	tmp = 0.0
	if (t <= -6e+59)
		tmp = t_1;
	elseif (t <= 2.7e+149)
		tmp = Float64(exp(Float64(Float64(-b) * a)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((-t * y)) * x;
	tmp = 0.0;
	if (t <= -6e+59)
		tmp = t_1;
	elseif (t <= 2.7e+149)
		tmp = exp((-b * a)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -6e+59], t$95$1, If[LessEqual[t, 2.7e+149], N[(N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.0000000000000001e59 or 2.7000000000000001e149 < t

   1. Initial program 98.7%

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

   3. Taylor expanded in 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 87.9

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

   5. Applied rewrites 87.9%

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

   if -6.0000000000000001e59 < t < 2.7000000000000001e149

   1. Initial program 95.0%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 66.8

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

   5. Applied rewrites 66.8%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+59}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+149}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 58.0

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

5. Applied rewrites 58.0%

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

6. Final simplification 58.0%

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

Alternative 7: 28.2% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, x, \left(\left(\left(-t\right) \cdot y\right) \cdot x\right) \cdot 1\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

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

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

   7. lower-*.f64 N/A

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

5. Applied rewrites 65.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 38.2%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 23.2%

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

13. Applied rewrites 24.0%

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

14. Final simplification 24.0%

    \[\leadsto \mathsf{fma}\left(1, x, \left(\left(\left(-t\right) \cdot y\right) \cdot x\right) \cdot 1\right) \]
15. Add Preprocessing

Alternative 8: 26.7% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \left(x - \left(t \cdot x\right) \cdot y\right) \cdot 1 \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x - ((t * x) * y)) * 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 - ((t * x) * y)) * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

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

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

   7. lower-*.f64 N/A

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

5. Applied rewrites 65.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 38.2%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 23.2%

    \[\leadsto 1 \cdot \left(x - \left(x \cdot t\right) \cdot y\right) \]

12. Final simplification 23.2%

    \[\leadsto \left(x - \left(t \cdot x\right) \cdot y\right) \cdot 1 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(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))))))