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

Percentage Accurate: 96.3% → 99.1%

Time: 18.1s

Alternatives: 10

Speedup: 1.5×

• 40.6% 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: 96.3% 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: 99.1% accurate, 1.5× speedup

Math

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

FPCore

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

Python

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

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) - (a * (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[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in z around 0 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-*r* 100.0%

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

   3. associate-*r* 100.0%

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

   4. distribute-lft-out 100.0%

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

   5. neg-mul-1 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 95.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) - (a * 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 * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in z around 0 97.7%

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

   1. +-commutative 97.7%

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

   2. mul-1-neg 97.7%

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

   3. sub-neg 97.7%

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

5. Simplified 97.7%

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

6. Final simplification 97.7%

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

Alternative 3: 84.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in z around 0 97.7%

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

   1. +-commutative 97.7%

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

   2. mul-1-neg 97.7%

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

   3. sub-neg 97.7%

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

5. Simplified 97.7%

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

6. Taylor expanded in t around inf 88.0%

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

   1. mul-1-neg 57.1%

      \[\leadsto x \cdot e^{\color{blue}{-t \cdot y}} \]

   2. distribute-lft-neg-out 57.1%

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

   3. *-commutative 57.1%

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

8. Simplified 88.0%

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

9. Final simplification 88.0%

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

Alternative 4: 63.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in z around 0 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-*r* 100.0%

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

   3. associate-*r* 100.0%

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

   4. distribute-lft-out 100.0%

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

   5. neg-mul-1 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in y around 0 62.9%

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

   1. *-commutative 62.9%

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

   2. mul-1-neg 62.9%

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

   3. distribute-rgt-neg-in 62.9%

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

8. Simplified 62.9%

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

9. Final simplification 62.9%

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

Alternative 5: 58.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in y around 0 61.4%

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

4. Taylor expanded in z around 0 61.0%

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

   1. neg-mul-1 61.0%

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

   2. distribute-rgt-neg-in 61.0%

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

6. Simplified 61.0%

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

7. Final simplification 61.0%

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

Alternative 6: 52.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x \cdot {z}^{y} \end{array} \]

FPCore

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

C

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

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 * (z ** y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in a around 0 73.1%

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

4. Taylor expanded in t around 0 47.6%

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

5. Final simplification 47.6%

   \[\leadsto x \cdot {z}^{y} \]
6. Add Preprocessing

Alternative 7: 27.2% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in z around 0 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-*r* 100.0%

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

   3. associate-*r* 100.0%

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

   4. distribute-lft-out 100.0%

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

   5. neg-mul-1 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in y around 0 62.9%

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

   1. *-commutative 62.9%

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

   2. mul-1-neg 62.9%

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

   3. distribute-rgt-neg-in 62.9%

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

8. Simplified 62.9%

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

9. Taylor expanded in a around 0 23.7%

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

    1. mul-1-neg 23.7%

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

    2. unsub-neg 23.7%

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

    3. +-commutative 23.7%

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

11. Simplified 23.7%

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

12. Final simplification 23.7%

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

Alternative 8: 28.5% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in z around 0 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-*r* 100.0%

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

   3. associate-*r* 100.0%

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

   4. distribute-lft-out 100.0%

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

   5. neg-mul-1 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in y around 0 62.9%

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

   1. *-commutative 62.9%

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

   2. mul-1-neg 62.9%

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

   3. distribute-rgt-neg-in 62.9%

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

8. Simplified 62.9%

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

9. Taylor expanded in a around 0 25.9%

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

    1. associate-*r* 25.9%

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

    2. neg-mul-1 25.9%

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

    3. +-commutative 25.9%

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

11. Simplified 25.9%

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

12. Final simplification 25.9%

    \[\leadsto x \cdot \left(1 - a \cdot \left(z + b\right)\right) \]
13. Add Preprocessing

Alternative 9: 28.6% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

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 - (y * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in a around 0 73.1%

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

4. Taylor expanded in t around inf 57.1%

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

   1. mul-1-neg 57.1%

      \[\leadsto x \cdot e^{\color{blue}{-t \cdot y}} \]

   2. distribute-lft-neg-out 57.1%

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

   3. *-commutative 57.1%

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

6. Simplified 57.1%

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

7. Taylor expanded in y around 0 22.9%

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

   1. mul-1-neg 22.9%

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

   2. *-commutative 22.9%

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

   3. unsub-neg 22.9%

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

9. Simplified 22.9%

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

10. Final simplification 22.9%

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

Alternative 10: 19.6% accurate, 315.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in a around 0 73.1%

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

4. Taylor expanded in y around 0 15.7%

   \[\leadsto \color{blue}{x} \]

5. Final simplification 15.7%

   \[\leadsto x \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024033 
(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))))))