Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A

Percentage Accurate: 99.3% → 99.8%

Time: 26.6s

Alternatives: 10

Speedup: 1.0×

• 43.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * exp(((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.exp(((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.exp(((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * exp(Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * exp(((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.exp(((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.exp(((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * exp(Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \sqrt{2 \cdot \left(z \cdot {\left(e^{2 \cdot t}\right)}^{\left(t \cdot 0.5\right)}\right)} \cdot \left(0.5 \cdot x - y\right) \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (* (sqrt (* 2.0 (* z (pow (exp (* 2.0 t)) (* t 0.5))))) (- (* 0.5 x) y)))

C

t = abs(t);
double code(double x, double y, double z, double t) {
	return sqrt((2.0 * (z * pow(exp((2.0 * t)), (t * 0.5))))) * ((0.5 * x) - y);
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt((2.0d0 * (z * (exp((2.0d0 * t)) ** (t * 0.5d0))))) * ((0.5d0 * x) - y)
end function

Java

t = Math.abs(t);
public static double code(double x, double y, double z, double t) {
	return Math.sqrt((2.0 * (z * Math.pow(Math.exp((2.0 * t)), (t * 0.5))))) * ((0.5 * x) - y);
}

Python

t = abs(t)
def code(x, y, z, t):
	return math.sqrt((2.0 * (z * math.pow(math.exp((2.0 * t)), (t * 0.5))))) * ((0.5 * x) - y)

Julia

t = abs(t)
function code(x, y, z, t)
	return Float64(sqrt(Float64(2.0 * Float64(z * (exp(Float64(2.0 * t)) ^ Float64(t * 0.5))))) * Float64(Float64(0.5 * x) - y))
end

MATLAB

t = abs(t)
function tmp = code(x, y, z, t)
	tmp = sqrt((2.0 * (z * (exp((2.0 * t)) ^ (t * 0.5))))) * ((0.5 * x) - y);
end

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := N[(N[Sqrt[N[(2.0 * N[(z * N[Power[N[Exp[N[(2.0 * t), $MachinePrecision]], $MachinePrecision], N[(t * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. associate-*l* 99.7%

      \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)} \]

   2. exp-sqrt 99.7%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right) \]

   3. exp-prod 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}}\right) \]

   4. *-commutative 99.8%

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(x \cdot 0.5 - y\right)} \]

   5. sub-neg 99.8%

      \[\leadsto \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \color{blue}{\left(x \cdot 0.5 + \left(-y\right)\right)} \]

   6. distribute-lft-in 80.7%

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right)} \]

   7. sqrt-unprod 80.7%

      \[\leadsto \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot {\left(e^{t}\right)}^{t}}} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

   8. associate-*l* 80.7%

      \[\leadsto \sqrt{\color{blue}{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

   9. exp-prod 80.6%

      \[\leadsto \sqrt{z \cdot \left(2 \cdot \color{blue}{e^{t \cdot t}}\right)} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

   10. pow2 80.6%

       \[\leadsto \sqrt{z \cdot \left(2 \cdot e^{\color{blue}{{t}^{2}}}\right)} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

3. Applied egg-rr 80.6%

   \[\leadsto \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(x \cdot 0.5\right) + \sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(-y\right)} \]
4. Step-by-step derivation

   1. distribute-lft-out 99.7%

      \[\leadsto \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(x \cdot 0.5 + \left(-y\right)\right)} \]

   2. sub-neg 99.7%

      \[\leadsto \sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \color{blue}{\left(x \cdot 0.5 - y\right)} \]

   3. *-commutative 99.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot e^{{t}^{2}}\right) \cdot z}} \cdot \left(x \cdot 0.5 - y\right) \]

   4. associate-*l* 99.7%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(e^{{t}^{2}} \cdot z\right)}} \cdot \left(x \cdot 0.5 - y\right) \]

   5. *-commutative 99.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(z \cdot e^{{t}^{2}}\right)}} \cdot \left(x \cdot 0.5 - y\right) \]

   6. *-commutative 99.7%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \cdot \left(\color{blue}{0.5 \cdot x} - y\right) \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \cdot \left(0.5 \cdot x - y\right)} \]
6. Step-by-step derivation

   1. unpow2 99.7%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot e^{\color{blue}{t \cdot t}}\right)} \cdot \left(0.5 \cdot x - y\right) \]

   2. exp-prod 99.8%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot \color{blue}{{\left(e^{t}\right)}^{t}}\right)} \cdot \left(0.5 \cdot x - y\right) \]

   3. pow1 99.8%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot \color{blue}{{\left({\left(e^{t}\right)}^{t}\right)}^{1}}\right)} \cdot \left(0.5 \cdot x - y\right) \]

   4. metadata-eval 99.8%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot {\left({\left(e^{t}\right)}^{t}\right)}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right)} \cdot \left(0.5 \cdot x - y\right) \]

   5. pow-sqr 99.8%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot \color{blue}{\left({\left({\left(e^{t}\right)}^{t}\right)}^{0.5} \cdot {\left({\left(e^{t}\right)}^{t}\right)}^{0.5}\right)}\right)} \cdot \left(0.5 \cdot x - y\right) \]

   6. pow-unpow 99.8%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot \left(\color{blue}{{\left(e^{t}\right)}^{\left(t \cdot 0.5\right)}} \cdot {\left({\left(e^{t}\right)}^{t}\right)}^{0.5}\right)\right)} \cdot \left(0.5 \cdot x - y\right) \]

   7. pow-unpow 99.8%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot \left({\left(e^{t}\right)}^{\left(t \cdot 0.5\right)} \cdot \color{blue}{{\left(e^{t}\right)}^{\left(t \cdot 0.5\right)}}\right)\right)} \cdot \left(0.5 \cdot x - y\right) \]

   8. pow-prod-down 99.8%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot \color{blue}{{\left(e^{t} \cdot e^{t}\right)}^{\left(t \cdot 0.5\right)}}\right)} \cdot \left(0.5 \cdot x - y\right) \]

   9. pow2 99.8%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot {\color{blue}{\left({\left(e^{t}\right)}^{2}\right)}}^{\left(t \cdot 0.5\right)}\right)} \cdot \left(0.5 \cdot x - y\right) \]

7. Applied egg-rr 99.8%

   \[\leadsto \sqrt{2 \cdot \left(z \cdot \color{blue}{{\left({\left(e^{t}\right)}^{2}\right)}^{\left(t \cdot 0.5\right)}}\right)} \cdot \left(0.5 \cdot x - y\right) \]
8. Step-by-step derivation

   1. pow-exp 99.8%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot {\color{blue}{\left(e^{t \cdot 2}\right)}}^{\left(t \cdot 0.5\right)}\right)} \cdot \left(0.5 \cdot x - y\right) \]

   2. *-commutative 99.8%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot {\left(e^{\color{blue}{2 \cdot t}}\right)}^{\left(t \cdot 0.5\right)}\right)} \cdot \left(0.5 \cdot x - y\right) \]

9. Applied egg-rr 99.8%

   \[\leadsto \sqrt{2 \cdot \left(z \cdot {\color{blue}{\left(e^{2 \cdot t}\right)}}^{\left(t \cdot 0.5\right)}\right)} \cdot \left(0.5 \cdot x - y\right) \]

10. Final simplification 99.8%

    \[\leadsto \sqrt{2 \cdot \left(z \cdot {\left(e^{2 \cdot t}\right)}^{\left(t \cdot 0.5\right)}\right)} \cdot \left(0.5 \cdot x - y\right) \]

Alternative 2: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (* (- (* 0.5 x) y) (sqrt (* 2.0 (* z (exp (pow t 2.0)))))))

C

t = abs(t);
double code(double x, double y, double z, double t) {
	return ((0.5 * x) - y) * sqrt((2.0 * (z * exp(pow(t, 2.0)))));
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((0.5d0 * x) - y) * sqrt((2.0d0 * (z * exp((t ** 2.0d0)))))
end function

Java

t = Math.abs(t);
public static double code(double x, double y, double z, double t) {
	return ((0.5 * x) - y) * Math.sqrt((2.0 * (z * Math.exp(Math.pow(t, 2.0)))));
}

Python

t = abs(t)
def code(x, y, z, t):
	return ((0.5 * x) - y) * math.sqrt((2.0 * (z * math.exp(math.pow(t, 2.0)))))

Julia

t = abs(t)
function code(x, y, z, t)
	return Float64(Float64(Float64(0.5 * x) - y) * sqrt(Float64(2.0 * Float64(z * exp((t ^ 2.0))))))
end

MATLAB

t = abs(t)
function tmp = code(x, y, z, t)
	tmp = ((0.5 * x) - y) * sqrt((2.0 * (z * exp((t ^ 2.0)))));
end

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(z * N[Exp[N[Power[t, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. associate-*l* 99.7%

      \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)} \]

   2. exp-sqrt 99.7%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right) \]

   3. exp-prod 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}}\right) \]

   4. *-commutative 99.8%

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(x \cdot 0.5 - y\right)} \]

   5. sub-neg 99.8%

      \[\leadsto \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \color{blue}{\left(x \cdot 0.5 + \left(-y\right)\right)} \]

   6. distribute-lft-in 80.7%

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right)} \]

   7. sqrt-unprod 80.7%

      \[\leadsto \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot {\left(e^{t}\right)}^{t}}} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

   8. associate-*l* 80.7%

      \[\leadsto \sqrt{\color{blue}{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

   9. exp-prod 80.6%

      \[\leadsto \sqrt{z \cdot \left(2 \cdot \color{blue}{e^{t \cdot t}}\right)} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

   10. pow2 80.6%

       \[\leadsto \sqrt{z \cdot \left(2 \cdot e^{\color{blue}{{t}^{2}}}\right)} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

3. Applied egg-rr 80.6%

   \[\leadsto \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(x \cdot 0.5\right) + \sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(-y\right)} \]
4. Step-by-step derivation

   1. distribute-lft-out 99.7%

      \[\leadsto \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(x \cdot 0.5 + \left(-y\right)\right)} \]

   2. sub-neg 99.7%

      \[\leadsto \sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \color{blue}{\left(x \cdot 0.5 - y\right)} \]

   3. *-commutative 99.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot e^{{t}^{2}}\right) \cdot z}} \cdot \left(x \cdot 0.5 - y\right) \]

   4. associate-*l* 99.7%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(e^{{t}^{2}} \cdot z\right)}} \cdot \left(x \cdot 0.5 - y\right) \]

   5. *-commutative 99.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(z \cdot e^{{t}^{2}}\right)}} \cdot \left(x \cdot 0.5 - y\right) \]

   6. *-commutative 99.7%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \cdot \left(\color{blue}{0.5 \cdot x} - y\right) \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \cdot \left(0.5 \cdot x - y\right)} \]

6. Final simplification 99.7%

   \[\leadsto \left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \]

Alternative 3: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot \left(z \cdot {\left(e^{t}\right)}^{t}\right)} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (* (- (* 0.5 x) y) (sqrt (* 2.0 (* z (pow (exp t) t))))))

C

t = abs(t);
double code(double x, double y, double z, double t) {
	return ((0.5 * x) - y) * sqrt((2.0 * (z * pow(exp(t), t))));
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((0.5d0 * x) - y) * sqrt((2.0d0 * (z * (exp(t) ** t))))
end function

Java

t = Math.abs(t);
public static double code(double x, double y, double z, double t) {
	return ((0.5 * x) - y) * Math.sqrt((2.0 * (z * Math.pow(Math.exp(t), t))));
}

Python

t = abs(t)
def code(x, y, z, t):
	return ((0.5 * x) - y) * math.sqrt((2.0 * (z * math.pow(math.exp(t), t))))

Julia

t = abs(t)
function code(x, y, z, t)
	return Float64(Float64(Float64(0.5 * x) - y) * sqrt(Float64(2.0 * Float64(z * (exp(t) ^ t)))))
end

MATLAB

t = abs(t)
function tmp = code(x, y, z, t)
	tmp = ((0.5 * x) - y) * sqrt((2.0 * (z * (exp(t) ^ t))));
end

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(z * N[Power[N[Exp[t], $MachinePrecision], t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. associate-*l* 99.7%

      \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)} \]

   2. exp-sqrt 99.7%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right) \]

   3. exp-prod 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}}\right) \]

   4. *-commutative 99.8%

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(x \cdot 0.5 - y\right)} \]

   5. sub-neg 99.8%

      \[\leadsto \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \color{blue}{\left(x \cdot 0.5 + \left(-y\right)\right)} \]

   6. distribute-lft-in 80.7%

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right)} \]

   7. sqrt-unprod 80.7%

      \[\leadsto \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot {\left(e^{t}\right)}^{t}}} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

   8. associate-*l* 80.7%

      \[\leadsto \sqrt{\color{blue}{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

   9. exp-prod 80.6%

      \[\leadsto \sqrt{z \cdot \left(2 \cdot \color{blue}{e^{t \cdot t}}\right)} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

   10. pow2 80.6%

       \[\leadsto \sqrt{z \cdot \left(2 \cdot e^{\color{blue}{{t}^{2}}}\right)} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

3. Applied egg-rr 80.6%

   \[\leadsto \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(x \cdot 0.5\right) + \sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(-y\right)} \]
4. Step-by-step derivation

   1. distribute-lft-out 99.7%

      \[\leadsto \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(x \cdot 0.5 + \left(-y\right)\right)} \]

   2. sub-neg 99.7%

      \[\leadsto \sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \color{blue}{\left(x \cdot 0.5 - y\right)} \]

   3. *-commutative 99.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot e^{{t}^{2}}\right) \cdot z}} \cdot \left(x \cdot 0.5 - y\right) \]

   4. associate-*l* 99.7%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(e^{{t}^{2}} \cdot z\right)}} \cdot \left(x \cdot 0.5 - y\right) \]

   5. *-commutative 99.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(z \cdot e^{{t}^{2}}\right)}} \cdot \left(x \cdot 0.5 - y\right) \]

   6. *-commutative 99.7%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \cdot \left(\color{blue}{0.5 \cdot x} - y\right) \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \cdot \left(0.5 \cdot x - y\right)} \]
6. Step-by-step derivation

   1. unpow2 99.7%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot e^{\color{blue}{t \cdot t}}\right)} \cdot \left(0.5 \cdot x - y\right) \]

   2. exp-prod 99.8%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot \color{blue}{{\left(e^{t}\right)}^{t}}\right)} \cdot \left(0.5 \cdot x - y\right) \]

7. Applied egg-rr 99.8%

   \[\leadsto \sqrt{2 \cdot \left(z \cdot \color{blue}{{\left(e^{t}\right)}^{t}}\right)} \cdot \left(0.5 \cdot x - y\right) \]

8. Final simplification 99.8%

   \[\leadsto \left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot \left(z \cdot {\left(e^{t}\right)}^{t}\right)} \]

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ e^{\frac{t \cdot t}{2}} \cdot \left(\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\right) \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (* (exp (/ (* t t) 2.0)) (* (- (* 0.5 x) y) (sqrt (* 2.0 z)))))

C

t = abs(t);
double code(double x, double y, double z, double t) {
	return exp(((t * t) / 2.0)) * (((0.5 * x) - y) * sqrt((2.0 * z)));
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = exp(((t * t) / 2.0d0)) * (((0.5d0 * x) - y) * sqrt((2.0d0 * z)))
end function

Java

t = Math.abs(t);
public static double code(double x, double y, double z, double t) {
	return Math.exp(((t * t) / 2.0)) * (((0.5 * x) - y) * Math.sqrt((2.0 * z)));
}

Python

t = abs(t)
def code(x, y, z, t):
	return math.exp(((t * t) / 2.0)) * (((0.5 * x) - y) * math.sqrt((2.0 * z)))

Julia

t = abs(t)
function code(x, y, z, t)
	return Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(Float64(Float64(0.5 * x) - y) * sqrt(Float64(2.0 * z))))
end

MATLAB

t = abs(t)
function tmp = code(x, y, z, t)
	tmp = exp(((t * t) / 2.0)) * (((0.5 * x) - y) * sqrt((2.0 * z)));
end

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

2. Final simplification 99.4%

   \[\leadsto e^{\frac{t \cdot t}{2}} \cdot \left(\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\right) \]

Alternative 5: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} t_1 := 0.5 \cdot x - y\\ \mathbf{if}\;t \leq 55:\\ \;\;\;\;t_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+46} \lor \neg \left(t \leq 2 \cdot 10^{+119}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \left(t_1 \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\left(2 \cdot z\right) \cdot {y}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* 0.5 x) y)))
   (if (<= t 55.0)
     (* t_1 (sqrt (* 2.0 z)))
     (if (or (<= t 3.4e+46) (not (<= t 2e+119)))
       (sqrt (* (* 2.0 z) (* t_1 t_1)))
       (- (sqrt (* (* 2.0 z) (pow y 2.0))))))))

C

t = abs(t);
double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double tmp;
	if (t <= 55.0) {
		tmp = t_1 * sqrt((2.0 * z));
	} else if ((t <= 3.4e+46) || !(t <= 2e+119)) {
		tmp = sqrt(((2.0 * z) * (t_1 * t_1)));
	} else {
		tmp = -sqrt(((2.0 * z) * pow(y, 2.0)));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (0.5d0 * x) - y
    if (t <= 55.0d0) then
        tmp = t_1 * sqrt((2.0d0 * z))
    else if ((t <= 3.4d+46) .or. (.not. (t <= 2d+119))) then
        tmp = sqrt(((2.0d0 * z) * (t_1 * t_1)))
    else
        tmp = -sqrt(((2.0d0 * z) * (y ** 2.0d0)))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double tmp;
	if (t <= 55.0) {
		tmp = t_1 * Math.sqrt((2.0 * z));
	} else if ((t <= 3.4e+46) || !(t <= 2e+119)) {
		tmp = Math.sqrt(((2.0 * z) * (t_1 * t_1)));
	} else {
		tmp = -Math.sqrt(((2.0 * z) * Math.pow(y, 2.0)));
	}
	return tmp;
}

Python

t = abs(t)
def code(x, y, z, t):
	t_1 = (0.5 * x) - y
	tmp = 0
	if t <= 55.0:
		tmp = t_1 * math.sqrt((2.0 * z))
	elif (t <= 3.4e+46) or not (t <= 2e+119):
		tmp = math.sqrt(((2.0 * z) * (t_1 * t_1)))
	else:
		tmp = -math.sqrt(((2.0 * z) * math.pow(y, 2.0)))
	return tmp

Julia

t = abs(t)
function code(x, y, z, t)
	t_1 = Float64(Float64(0.5 * x) - y)
	tmp = 0.0
	if (t <= 55.0)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	elseif ((t <= 3.4e+46) || !(t <= 2e+119))
		tmp = sqrt(Float64(Float64(2.0 * z) * Float64(t_1 * t_1)));
	else
		tmp = Float64(-sqrt(Float64(Float64(2.0 * z) * (y ^ 2.0))));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(x, y, z, t)
	t_1 = (0.5 * x) - y;
	tmp = 0.0;
	if (t <= 55.0)
		tmp = t_1 * sqrt((2.0 * z));
	elseif ((t <= 3.4e+46) || ~((t <= 2e+119)))
		tmp = sqrt(((2.0 * z) * (t_1 * t_1)));
	else
		tmp = -sqrt(((2.0 * z) * (y ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 55.0], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 3.4e+46], N[Not[LessEqual[t, 2e+119]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], (-N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if t < 55

   1. Initial program 99.2%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Step-by-step derivation

      1. associate-*l* 99.7%

         \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)} \]

      2. exp-sqrt 99.7%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right) \]

      3. exp-prod 99.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}}\right) \]

      4. *-commutative 99.8%

         \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(x \cdot 0.5 - y\right)} \]

      5. sub-neg 99.8%

         \[\leadsto \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \color{blue}{\left(x \cdot 0.5 + \left(-y\right)\right)} \]

      6. distribute-lft-in 87.3%

         \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right)} \]

      7. sqrt-unprod 87.3%

         \[\leadsto \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot {\left(e^{t}\right)}^{t}}} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

      8. associate-*l* 87.3%

         \[\leadsto \sqrt{\color{blue}{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

      9. exp-prod 87.2%

         \[\leadsto \sqrt{z \cdot \left(2 \cdot \color{blue}{e^{t \cdot t}}\right)} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

      10. pow2 87.2%

          \[\leadsto \sqrt{z \cdot \left(2 \cdot e^{\color{blue}{{t}^{2}}}\right)} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

   3. Applied egg-rr 87.2%

      \[\leadsto \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(x \cdot 0.5\right) + \sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(-y\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-out 99.7%

         \[\leadsto \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(x \cdot 0.5 + \left(-y\right)\right)} \]

      2. sub-neg 99.7%

         \[\leadsto \sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \color{blue}{\left(x \cdot 0.5 - y\right)} \]

      3. *-commutative 99.7%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot e^{{t}^{2}}\right) \cdot z}} \cdot \left(x \cdot 0.5 - y\right) \]

      4. associate-*l* 99.7%

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(e^{{t}^{2}} \cdot z\right)}} \cdot \left(x \cdot 0.5 - y\right) \]

      5. *-commutative 99.7%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(z \cdot e^{{t}^{2}}\right)}} \cdot \left(x \cdot 0.5 - y\right) \]

      6. *-commutative 99.7%

         \[\leadsto \sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \cdot \left(\color{blue}{0.5 \cdot x} - y\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \cdot \left(0.5 \cdot x - y\right)} \]

   6. Taylor expanded in t around 0 68.4%

      \[\leadsto \sqrt{2 \cdot \color{blue}{z}} \cdot \left(0.5 \cdot x - y\right) \]

   if 55 < t < 3.3999999999999998e46 or 1.99999999999999989e119 < t

   1. Initial program 100.0%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in t around 0 22.5%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

      1. add-sqr-sqrt 13.3%

         \[\leadsto \color{blue}{\left(\sqrt{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot \sqrt{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)} \cdot 1 \]

      2. sqrt-unprod 35.3%

         \[\leadsto \color{blue}{\sqrt{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right)}} \cdot 1 \]

      3. *-commutative 35.3%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right)} \cdot 1 \]

      4. *-commutative 35.3%

         \[\leadsto \sqrt{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)}} \cdot 1 \]

      5. swap-sqr 37.8%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)}} \cdot 1 \]

      6. add-sqr-sqrt 37.8%

         \[\leadsto \sqrt{\color{blue}{\left(z \cdot 2\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)} \cdot 1 \]

      7. pow2 37.8%

         \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \color{blue}{{\left(x \cdot 0.5 - y\right)}^{2}}} \cdot 1 \]

   4. Applied egg-rr 37.8%

      \[\leadsto \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}} \cdot 1 \]
   5. Step-by-step derivation

      1. *-commutative 37.8%

         \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot {\left(\color{blue}{0.5 \cdot x} - y\right)}^{2}} \cdot 1 \]

      2. unpow2 37.8%

         \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \color{blue}{\left(\left(0.5 \cdot x - y\right) \cdot \left(0.5 \cdot x - y\right)\right)}} \cdot 1 \]

   6. Applied egg-rr 37.8%

      \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \color{blue}{\left(\left(0.5 \cdot x - y\right) \cdot \left(0.5 \cdot x - y\right)\right)}} \cdot 1 \]

   if 3.3999999999999998e46 < t < 1.99999999999999989e119

   1. Initial program 100.0%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in t around 0 4.8%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{1} \]

   3. Taylor expanded in x around 0 4.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \cdot 1 \]
   4. Step-by-step derivation

      1. mul-1-neg 4.4%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot 1 \]

      2. associate-*l* 4.4%

         \[\leadsto \left(-\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}\right) \cdot 1 \]

   5. Simplified 4.4%

      \[\leadsto \color{blue}{\left(-y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right)} \cdot 1 \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 3.5%

         \[\leadsto \left(-\color{blue}{\sqrt{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \cdot \sqrt{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}}\right) \cdot 1 \]

      2. sqrt-unprod 29.3%

         \[\leadsto \left(-\color{blue}{\sqrt{\left(y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right) \cdot \left(y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right)}}\right) \cdot 1 \]

      3. *-commutative 29.3%

         \[\leadsto \left(-\sqrt{\color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{z}\right) \cdot y\right)} \cdot \left(y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right)}\right) \cdot 1 \]

      4. *-commutative 29.3%

         \[\leadsto \left(-\sqrt{\left(\left(\sqrt{2} \cdot \sqrt{z}\right) \cdot y\right) \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{z}\right) \cdot y\right)}}\right) \cdot 1 \]

      5. swap-sqr 39.9%

         \[\leadsto \left(-\sqrt{\color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{z}\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right) \cdot \left(y \cdot y\right)}}\right) \cdot 1 \]

      6. sqrt-prod 39.9%

         \[\leadsto \left(-\sqrt{\left(\color{blue}{\sqrt{2 \cdot z}} \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right) \cdot \left(y \cdot y\right)}\right) \cdot 1 \]

      7. sqrt-prod 39.9%

         \[\leadsto \left(-\sqrt{\left(\sqrt{2 \cdot z} \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot \left(y \cdot y\right)}\right) \cdot 1 \]

      8. add-sqr-sqrt 39.9%

         \[\leadsto \left(-\sqrt{\color{blue}{\left(2 \cdot z\right)} \cdot \left(y \cdot y\right)}\right) \cdot 1 \]

      9. pow2 39.9%

         \[\leadsto \left(-\sqrt{\left(2 \cdot z\right) \cdot \color{blue}{{y}^{2}}}\right) \cdot 1 \]

   7. Applied egg-rr 39.9%

      \[\leadsto \left(-\color{blue}{\sqrt{\left(2 \cdot z\right) \cdot {y}^{2}}}\right) \cdot 1 \]
   8. Step-by-step derivation

      1. *-commutative 39.9%

         \[\leadsto \left(-\sqrt{\color{blue}{{y}^{2} \cdot \left(2 \cdot z\right)}}\right) \cdot 1 \]

      2. *-commutative 39.9%

         \[\leadsto \left(-\sqrt{{y}^{2} \cdot \color{blue}{\left(z \cdot 2\right)}}\right) \cdot 1 \]

   9. Simplified 39.9%

      \[\leadsto \left(-\color{blue}{\sqrt{{y}^{2} \cdot \left(z \cdot 2\right)}}\right) \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 55:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+46} \lor \neg \left(t \leq 2 \cdot 10^{+119}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \left(\left(0.5 \cdot x - y\right) \cdot \left(0.5 \cdot x - y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\left(2 \cdot z\right) \cdot {y}^{2}}\\ \end{array} \]

Alternative 6: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot \left(z \cdot \mathsf{fma}\left(t, t, 1\right)\right)} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (* (- (* 0.5 x) y) (sqrt (* 2.0 (* z (fma t t 1.0))))))

C

t = abs(t);
double code(double x, double y, double z, double t) {
	return ((0.5 * x) - y) * sqrt((2.0 * (z * fma(t, t, 1.0))));
}

Julia

t = abs(t)
function code(x, y, z, t)
	return Float64(Float64(Float64(0.5 * x) - y) * sqrt(Float64(2.0 * Float64(z * fma(t, t, 1.0)))))
end

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(z * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. associate-*l* 99.7%

      \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)} \]

   2. exp-sqrt 99.7%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right) \]

   3. exp-prod 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}}\right) \]

   4. *-commutative 99.8%

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(x \cdot 0.5 - y\right)} \]

   5. sub-neg 99.8%

      \[\leadsto \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \color{blue}{\left(x \cdot 0.5 + \left(-y\right)\right)} \]

   6. distribute-lft-in 80.7%

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right)} \]

   7. sqrt-unprod 80.7%

      \[\leadsto \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot {\left(e^{t}\right)}^{t}}} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

   8. associate-*l* 80.7%

      \[\leadsto \sqrt{\color{blue}{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

   9. exp-prod 80.6%

      \[\leadsto \sqrt{z \cdot \left(2 \cdot \color{blue}{e^{t \cdot t}}\right)} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

   10. pow2 80.6%

       \[\leadsto \sqrt{z \cdot \left(2 \cdot e^{\color{blue}{{t}^{2}}}\right)} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

3. Applied egg-rr 80.6%

   \[\leadsto \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(x \cdot 0.5\right) + \sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(-y\right)} \]
4. Step-by-step derivation

   1. distribute-lft-out 99.7%

      \[\leadsto \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(x \cdot 0.5 + \left(-y\right)\right)} \]

   2. sub-neg 99.7%

      \[\leadsto \sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \color{blue}{\left(x \cdot 0.5 - y\right)} \]

   3. *-commutative 99.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot e^{{t}^{2}}\right) \cdot z}} \cdot \left(x \cdot 0.5 - y\right) \]

   4. associate-*l* 99.7%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(e^{{t}^{2}} \cdot z\right)}} \cdot \left(x \cdot 0.5 - y\right) \]

   5. *-commutative 99.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(z \cdot e^{{t}^{2}}\right)}} \cdot \left(x \cdot 0.5 - y\right) \]

   6. *-commutative 99.7%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \cdot \left(\color{blue}{0.5 \cdot x} - y\right) \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \cdot \left(0.5 \cdot x - y\right)} \]

6. Taylor expanded in t around 0 79.9%

   \[\leadsto \sqrt{2 \cdot \left(z \cdot \color{blue}{\left(1 + {t}^{2}\right)}\right)} \cdot \left(0.5 \cdot x - y\right) \]
7. Step-by-step derivation

   1. +-commutative 79.9%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot \color{blue}{\left({t}^{2} + 1\right)}\right)} \cdot \left(0.5 \cdot x - y\right) \]

   2. unpow2 79.9%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot \left(\color{blue}{t \cdot t} + 1\right)\right)} \cdot \left(0.5 \cdot x - y\right) \]

   3. fma-def 79.9%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot \color{blue}{\mathsf{fma}\left(t, t, 1\right)}\right)} \cdot \left(0.5 \cdot x - y\right) \]

8. Simplified 79.9%

   \[\leadsto \sqrt{2 \cdot \left(z \cdot \color{blue}{\mathsf{fma}\left(t, t, 1\right)}\right)} \cdot \left(0.5 \cdot x - y\right) \]

9. Final simplification 79.9%

   \[\leadsto \left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot \left(z \cdot \mathsf{fma}\left(t, t, 1\right)\right)} \]

Alternative 7: 62.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} t_1 := 0.5 \cdot x - y\\ \mathbf{if}\;t \leq 50:\\ \;\;\;\;t_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \left(t_1 \cdot t_1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* 0.5 x) y)))
   (if (<= t 50.0) (* t_1 (sqrt (* 2.0 z))) (sqrt (* (* 2.0 z) (* t_1 t_1))))))

C

t = abs(t);
double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double tmp;
	if (t <= 50.0) {
		tmp = t_1 * sqrt((2.0 * z));
	} else {
		tmp = sqrt(((2.0 * z) * (t_1 * t_1)));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (0.5d0 * x) - y
    if (t <= 50.0d0) then
        tmp = t_1 * sqrt((2.0d0 * z))
    else
        tmp = sqrt(((2.0d0 * z) * (t_1 * t_1)))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double tmp;
	if (t <= 50.0) {
		tmp = t_1 * Math.sqrt((2.0 * z));
	} else {
		tmp = Math.sqrt(((2.0 * z) * (t_1 * t_1)));
	}
	return tmp;
}

Python

t = abs(t)
def code(x, y, z, t):
	t_1 = (0.5 * x) - y
	tmp = 0
	if t <= 50.0:
		tmp = t_1 * math.sqrt((2.0 * z))
	else:
		tmp = math.sqrt(((2.0 * z) * (t_1 * t_1)))
	return tmp

Julia

t = abs(t)
function code(x, y, z, t)
	t_1 = Float64(Float64(0.5 * x) - y)
	tmp = 0.0
	if (t <= 50.0)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	else
		tmp = sqrt(Float64(Float64(2.0 * z) * Float64(t_1 * t_1)));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(x, y, z, t)
	t_1 = (0.5 * x) - y;
	tmp = 0.0;
	if (t <= 50.0)
		tmp = t_1 * sqrt((2.0 * z));
	else
		tmp = sqrt(((2.0 * z) * (t_1 * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 50.0], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 50

   1. Initial program 99.2%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Step-by-step derivation

      1. associate-*l* 99.7%

         \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)} \]

      2. exp-sqrt 99.7%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right) \]

      3. exp-prod 99.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}}\right) \]

      4. *-commutative 99.8%

         \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(x \cdot 0.5 - y\right)} \]

      5. sub-neg 99.8%

         \[\leadsto \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \color{blue}{\left(x \cdot 0.5 + \left(-y\right)\right)} \]

      6. distribute-lft-in 87.3%

         \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right)} \]

      7. sqrt-unprod 87.3%

         \[\leadsto \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot {\left(e^{t}\right)}^{t}}} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

      8. associate-*l* 87.3%

         \[\leadsto \sqrt{\color{blue}{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

      9. exp-prod 87.2%

         \[\leadsto \sqrt{z \cdot \left(2 \cdot \color{blue}{e^{t \cdot t}}\right)} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

      10. pow2 87.2%

          \[\leadsto \sqrt{z \cdot \left(2 \cdot e^{\color{blue}{{t}^{2}}}\right)} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

   3. Applied egg-rr 87.2%

      \[\leadsto \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(x \cdot 0.5\right) + \sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(-y\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-out 99.7%

         \[\leadsto \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(x \cdot 0.5 + \left(-y\right)\right)} \]

      2. sub-neg 99.7%

         \[\leadsto \sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \color{blue}{\left(x \cdot 0.5 - y\right)} \]

      3. *-commutative 99.7%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot e^{{t}^{2}}\right) \cdot z}} \cdot \left(x \cdot 0.5 - y\right) \]

      4. associate-*l* 99.7%

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(e^{{t}^{2}} \cdot z\right)}} \cdot \left(x \cdot 0.5 - y\right) \]

      5. *-commutative 99.7%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(z \cdot e^{{t}^{2}}\right)}} \cdot \left(x \cdot 0.5 - y\right) \]

      6. *-commutative 99.7%

         \[\leadsto \sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \cdot \left(\color{blue}{0.5 \cdot x} - y\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \cdot \left(0.5 \cdot x - y\right)} \]

   6. Taylor expanded in t around 0 68.4%

      \[\leadsto \sqrt{2 \cdot \color{blue}{z}} \cdot \left(0.5 \cdot x - y\right) \]

   if 50 < t

   1. Initial program 100.0%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in t around 0 16.8%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

      1. add-sqr-sqrt 9.4%

         \[\leadsto \color{blue}{\left(\sqrt{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot \sqrt{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)} \cdot 1 \]

      2. sqrt-unprod 26.1%

         \[\leadsto \color{blue}{\sqrt{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right)}} \cdot 1 \]

      3. *-commutative 26.1%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right)} \cdot 1 \]

      4. *-commutative 26.1%

         \[\leadsto \sqrt{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)}} \cdot 1 \]

      5. swap-sqr 29.5%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)}} \cdot 1 \]

      6. add-sqr-sqrt 29.5%

         \[\leadsto \sqrt{\color{blue}{\left(z \cdot 2\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)} \cdot 1 \]

      7. pow2 29.5%

         \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \color{blue}{{\left(x \cdot 0.5 - y\right)}^{2}}} \cdot 1 \]

   4. Applied egg-rr 29.5%

      \[\leadsto \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}} \cdot 1 \]
   5. Step-by-step derivation

      1. *-commutative 29.5%

         \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot {\left(\color{blue}{0.5 \cdot x} - y\right)}^{2}} \cdot 1 \]

      2. unpow2 29.5%

         \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \color{blue}{\left(\left(0.5 \cdot x - y\right) \cdot \left(0.5 \cdot x - y\right)\right)}} \cdot 1 \]

   6. Applied egg-rr 29.5%

      \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \color{blue}{\left(\left(0.5 \cdot x - y\right) \cdot \left(0.5 \cdot x - y\right)\right)}} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 50:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \left(\left(0.5 \cdot x - y\right) \cdot \left(0.5 \cdot x - y\right)\right)}\\ \end{array} \]

Alternative 8: 57.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 29:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= t 29.0)
   (* (- (* 0.5 x) y) (sqrt (* 2.0 z)))
   (sqrt (* (* 2.0 z) (* y (- y x))))))

C

t = abs(t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 29.0) {
		tmp = ((0.5 * x) - y) * sqrt((2.0 * z));
	} else {
		tmp = sqrt(((2.0 * z) * (y * (y - x))));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 29.0d0) then
        tmp = ((0.5d0 * x) - y) * sqrt((2.0d0 * z))
    else
        tmp = sqrt(((2.0d0 * z) * (y * (y - x))))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 29.0) {
		tmp = ((0.5 * x) - y) * Math.sqrt((2.0 * z));
	} else {
		tmp = Math.sqrt(((2.0 * z) * (y * (y - x))));
	}
	return tmp;
}

Python

t = abs(t)
def code(x, y, z, t):
	tmp = 0
	if t <= 29.0:
		tmp = ((0.5 * x) - y) * math.sqrt((2.0 * z))
	else:
		tmp = math.sqrt(((2.0 * z) * (y * (y - x))))
	return tmp

Julia

t = abs(t)
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 29.0)
		tmp = Float64(Float64(Float64(0.5 * x) - y) * sqrt(Float64(2.0 * z)));
	else
		tmp = sqrt(Float64(Float64(2.0 * z) * Float64(y * Float64(y - x))));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 29.0)
		tmp = ((0.5 * x) - y) * sqrt((2.0 * z));
	else
		tmp = sqrt(((2.0 * z) * (y * (y - x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[t, 29.0], N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 29

   1. Initial program 99.2%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Step-by-step derivation

      1. associate-*l* 99.7%

         \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)} \]

      2. exp-sqrt 99.7%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right) \]

      3. exp-prod 99.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}}\right) \]

      4. *-commutative 99.8%

         \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(x \cdot 0.5 - y\right)} \]

      5. sub-neg 99.8%

         \[\leadsto \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \color{blue}{\left(x \cdot 0.5 + \left(-y\right)\right)} \]

      6. distribute-lft-in 87.3%

         \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right)} \]

      7. sqrt-unprod 87.3%

         \[\leadsto \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot {\left(e^{t}\right)}^{t}}} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

      8. associate-*l* 87.3%

         \[\leadsto \sqrt{\color{blue}{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

      9. exp-prod 87.2%

         \[\leadsto \sqrt{z \cdot \left(2 \cdot \color{blue}{e^{t \cdot t}}\right)} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

      10. pow2 87.2%

          \[\leadsto \sqrt{z \cdot \left(2 \cdot e^{\color{blue}{{t}^{2}}}\right)} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

   3. Applied egg-rr 87.2%

      \[\leadsto \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(x \cdot 0.5\right) + \sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(-y\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-out 99.7%

         \[\leadsto \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(x \cdot 0.5 + \left(-y\right)\right)} \]

      2. sub-neg 99.7%

         \[\leadsto \sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \color{blue}{\left(x \cdot 0.5 - y\right)} \]

      3. *-commutative 99.7%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot e^{{t}^{2}}\right) \cdot z}} \cdot \left(x \cdot 0.5 - y\right) \]

      4. associate-*l* 99.7%

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(e^{{t}^{2}} \cdot z\right)}} \cdot \left(x \cdot 0.5 - y\right) \]

      5. *-commutative 99.7%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(z \cdot e^{{t}^{2}}\right)}} \cdot \left(x \cdot 0.5 - y\right) \]

      6. *-commutative 99.7%

         \[\leadsto \sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \cdot \left(\color{blue}{0.5 \cdot x} - y\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \cdot \left(0.5 \cdot x - y\right)} \]

   6. Taylor expanded in t around 0 68.4%

      \[\leadsto \sqrt{2 \cdot \color{blue}{z}} \cdot \left(0.5 \cdot x - y\right) \]

   if 29 < t

   1. Initial program 100.0%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in t around 0 16.8%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{1} \]
   3. Step-by-step derivation

      1. add-sqr-sqrt 9.4%

         \[\leadsto \color{blue}{\left(\sqrt{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot \sqrt{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)} \cdot 1 \]

      2. sqrt-unprod 26.1%

         \[\leadsto \color{blue}{\sqrt{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right)}} \cdot 1 \]

      3. *-commutative 26.1%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right)} \cdot 1 \]

      4. *-commutative 26.1%

         \[\leadsto \sqrt{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)}} \cdot 1 \]

      5. swap-sqr 29.5%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)}} \cdot 1 \]

      6. add-sqr-sqrt 29.5%

         \[\leadsto \sqrt{\color{blue}{\left(z \cdot 2\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)} \cdot 1 \]

      7. pow2 29.5%

         \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \color{blue}{{\left(x \cdot 0.5 - y\right)}^{2}}} \cdot 1 \]

   4. Applied egg-rr 29.5%

      \[\leadsto \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}} \cdot 1 \]

   5. Taylor expanded in x around 0 15.3%

      \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \color{blue}{\left(-1 \cdot \left(x \cdot y\right) + {y}^{2}\right)}} \cdot 1 \]
   6. Step-by-step derivation

      1. +-commutative 15.3%

         \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \color{blue}{\left({y}^{2} + -1 \cdot \left(x \cdot y\right)\right)}} \cdot 1 \]

      2. unpow2 15.3%

         \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \left(\color{blue}{y \cdot y} + -1 \cdot \left(x \cdot y\right)\right)} \cdot 1 \]

      3. associate-*r* 15.3%

         \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot y + \color{blue}{\left(-1 \cdot x\right) \cdot y}\right)} \cdot 1 \]

      4. distribute-rgt-out 18.8%

         \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \color{blue}{\left(y \cdot \left(y + -1 \cdot x\right)\right)}} \cdot 1 \]

      5. mul-1-neg 18.8%

         \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y + \color{blue}{\left(-x\right)}\right)\right)} \cdot 1 \]

   7. Simplified 18.8%

      \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \color{blue}{\left(y \cdot \left(y + \left(-x\right)\right)\right)}} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 29:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \end{array} \]

Alternative 9: 57.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t) :precision binary64 (* (- (* 0.5 x) y) (sqrt (* 2.0 z))))

C

t = abs(t);
double code(double x, double y, double z, double t) {
	return ((0.5 * x) - y) * sqrt((2.0 * z));
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((0.5d0 * x) - y) * sqrt((2.0d0 * z))
end function

Java

t = Math.abs(t);
public static double code(double x, double y, double z, double t) {
	return ((0.5 * x) - y) * Math.sqrt((2.0 * z));
}

Python

t = abs(t)
def code(x, y, z, t):
	return ((0.5 * x) - y) * math.sqrt((2.0 * z))

Julia

t = abs(t)
function code(x, y, z, t)
	return Float64(Float64(Float64(0.5 * x) - y) * sqrt(Float64(2.0 * z)))
end

MATLAB

t = abs(t)
function tmp = code(x, y, z, t)
	tmp = ((0.5 * x) - y) * sqrt((2.0 * z));
end

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. associate-*l* 99.7%

      \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)} \]

   2. exp-sqrt 99.7%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right) \]

   3. exp-prod 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}}\right) \]

   4. *-commutative 99.8%

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(x \cdot 0.5 - y\right)} \]

   5. sub-neg 99.8%

      \[\leadsto \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \color{blue}{\left(x \cdot 0.5 + \left(-y\right)\right)} \]

   6. distribute-lft-in 80.7%

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right)} \]

   7. sqrt-unprod 80.7%

      \[\leadsto \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot {\left(e^{t}\right)}^{t}}} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

   8. associate-*l* 80.7%

      \[\leadsto \sqrt{\color{blue}{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

   9. exp-prod 80.6%

      \[\leadsto \sqrt{z \cdot \left(2 \cdot \color{blue}{e^{t \cdot t}}\right)} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

   10. pow2 80.6%

       \[\leadsto \sqrt{z \cdot \left(2 \cdot e^{\color{blue}{{t}^{2}}}\right)} \cdot \left(x \cdot 0.5\right) + \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right) \cdot \left(-y\right) \]

3. Applied egg-rr 80.6%

   \[\leadsto \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(x \cdot 0.5\right) + \sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(-y\right)} \]
4. Step-by-step derivation

   1. distribute-lft-out 99.7%

      \[\leadsto \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \left(x \cdot 0.5 + \left(-y\right)\right)} \]

   2. sub-neg 99.7%

      \[\leadsto \sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)} \cdot \color{blue}{\left(x \cdot 0.5 - y\right)} \]

   3. *-commutative 99.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot e^{{t}^{2}}\right) \cdot z}} \cdot \left(x \cdot 0.5 - y\right) \]

   4. associate-*l* 99.7%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(e^{{t}^{2}} \cdot z\right)}} \cdot \left(x \cdot 0.5 - y\right) \]

   5. *-commutative 99.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(z \cdot e^{{t}^{2}}\right)}} \cdot \left(x \cdot 0.5 - y\right) \]

   6. *-commutative 99.7%

      \[\leadsto \sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \cdot \left(\color{blue}{0.5 \cdot x} - y\right) \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(z \cdot e^{{t}^{2}}\right)} \cdot \left(0.5 \cdot x - y\right)} \]

6. Taylor expanded in t around 0 57.1%

   \[\leadsto \sqrt{2 \cdot \color{blue}{z}} \cdot \left(0.5 \cdot x - y\right) \]

7. Final simplification 57.1%

   \[\leadsto \left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z} \]

Alternative 10: 30.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \sqrt{2 \cdot z} \cdot \left(-y\right) \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t) :precision binary64 (* (sqrt (* 2.0 z)) (- y)))

C

t = abs(t);
double code(double x, double y, double z, double t) {
	return sqrt((2.0 * z)) * -y;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt((2.0d0 * z)) * -y
end function

Java

t = Math.abs(t);
public static double code(double x, double y, double z, double t) {
	return Math.sqrt((2.0 * z)) * -y;
}

Python

t = abs(t)
def code(x, y, z, t):
	return math.sqrt((2.0 * z)) * -y

Julia

t = abs(t)
function code(x, y, z, t)
	return Float64(sqrt(Float64(2.0 * z)) * Float64(-y))
end

MATLAB

t = abs(t)
function tmp = code(x, y, z, t)
	tmp = sqrt((2.0 * z)) * -y;
end

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

2. Taylor expanded in t around 0 57.1%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{1} \]

3. Taylor expanded in x around 0 30.7%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \cdot 1 \]
4. Step-by-step derivation

   1. mul-1-neg 30.7%

      \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot 1 \]

   2. associate-*l* 30.8%

      \[\leadsto \left(-\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}\right) \cdot 1 \]

5. Simplified 30.8%

   \[\leadsto \color{blue}{\left(-y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right)} \cdot 1 \]
6. Step-by-step derivation

   1. expm1-log1p-u 30.2%

      \[\leadsto \left(-y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{z}\right)\right)}\right) \cdot 1 \]

   2. expm1-udef 17.7%

      \[\leadsto \left(-y \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{z}\right)} - 1\right)}\right) \cdot 1 \]

   3. sqrt-prod 17.7%

      \[\leadsto \left(-y \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{2 \cdot z}}\right)} - 1\right)\right) \cdot 1 \]

7. Applied egg-rr 17.7%

   \[\leadsto \left(-y \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot z}\right)} - 1\right)}\right) \cdot 1 \]
8. Step-by-step derivation

   1. expm1-def 30.2%

      \[\leadsto \left(-y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot z}\right)\right)}\right) \cdot 1 \]

   2. expm1-log1p 30.9%

      \[\leadsto \left(-y \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot 1 \]

   3. *-commutative 30.9%

      \[\leadsto \left(-y \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot 1 \]

9. Simplified 30.9%

   \[\leadsto \left(-y \cdot \color{blue}{\sqrt{z \cdot 2}}\right) \cdot 1 \]

10. Final simplification 30.9%

    \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]

Developer target: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{1}\right)}^{\left(\frac{t \cdot t}{2}\right)} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (exp 1.0) (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * pow(exp(1.0), ((t * t) / 2.0));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * (exp(1.0d0) ** ((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.pow(Math.exp(1.0), ((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.pow(math.exp(1.0), ((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * (exp(1.0) ^ Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * (exp(1.0) ^ ((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[1.0], $MachinePrecision], N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023301 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :herbie-target
  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (exp 1.0) (/ (* t t) 2.0)))

  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))