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

Percentage Accurate: 99.5% → 99.8%

Time: 13.6s

Alternatives: 8

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: 99.5% 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

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

FPCore

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

C

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

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 = sqrt((z * (2.0d0 * exp((t ** 2.0d0))))) * ((0.5d0 * x) - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. expm1-log1p-u 52.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\right)\right)} \]

   2. expm1-undefine 41.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\right)} - 1} \]

   3. associate-*l* 41.8%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)}\right)} - 1 \]

   4. exp-sqrt 41.8%

      \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right)\right)} - 1 \]

   5. sqrt-unprod 41.8%

      \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}}\right)} - 1 \]

   6. pow2 41.8%

      \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{\color{blue}{{t}^{2}}}}\right)} - 1 \]

4. Applied egg-rr 41.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{{t}^{2}}}\right)} - 1} \]
5. Step-by-step derivation

   1. log1p-undefine 41.8%

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

   2. rem-exp-log 88.9%

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

   3. +-commutative 88.9%

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

   4. associate--l+ 99.8%

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

   5. metadata-eval 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{{t}^{2}}} + \color{blue}{0} \]

   6. +-rgt-identity 99.8%

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

   7. *-commutative 99.8%

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

   8. associate-*l* 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 92.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot t \leq 0.0001 \lor \neg \left(t \cdot t \leq 5 \cdot 10^{+306}\right):\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot \left(z \cdot \mathsf{fma}\left(t, t, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(-\sqrt{z \cdot 2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* t t) 0.0001) (not (<= (* t t) 5e+306)))
   (* (- (* 0.5 x) y) (sqrt (* 2.0 (* z (fma t t 1.0)))))
   (* (exp (/ (* t t) 2.0)) (* y (- (sqrt (* z 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((t * t) <= 0.0001) || !((t * t) <= 5e+306)) {
		tmp = ((0.5 * x) - y) * sqrt((2.0 * (z * fma(t, t, 1.0))));
	} else {
		tmp = exp(((t * t) / 2.0)) * (y * -sqrt((z * 2.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(t * t) <= 0.0001) || !(Float64(t * t) <= 5e+306))
		tmp = Float64(Float64(Float64(0.5 * x) - y) * sqrt(Float64(2.0 * Float64(z * fma(t, t, 1.0)))));
	else
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(y * Float64(-sqrt(Float64(z * 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(t * t), $MachinePrecision], 0.0001], N[Not[LessEqual[N[(t * t), $MachinePrecision], 5e+306]], $MachinePrecision]], 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], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(y * (-N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 1.00000000000000005e-4 or 4.99999999999999993e306 < (*.f64 t t)

   1. Initial program 99.8%

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

      1. expm1-log1p-u 54.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\right)\right)} \]

      2. expm1-undefine 39.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\right)} - 1} \]

      3. associate-*l* 39.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)}\right)} - 1 \]

      4. exp-sqrt 39.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right)\right)} - 1 \]

      5. sqrt-unprod 39.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}}\right)} - 1 \]

      6. pow2 39.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{\color{blue}{{t}^{2}}}}\right)} - 1 \]

   4. Applied egg-rr 39.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{{t}^{2}}}\right)} - 1} \]
   5. Step-by-step derivation

      1. log1p-undefine 39.9%

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

      2. rem-exp-log 84.9%

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

      3. +-commutative 84.9%

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

      4. associate--l+ 99.8%

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

      5. metadata-eval 99.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{{t}^{2}}} + \color{blue}{0} \]

      6. +-rgt-identity 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l* 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. *-commutative 99.6%

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

   9. Simplified 99.6%

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

       1. pow1 99.6%

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

       2. *-commutative 99.6%

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

       3. +-commutative 99.6%

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

       4. pow2 99.6%

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

       5. fma-define 99.6%

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

       6. pow2 99.6%

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

   11. Applied egg-rr 99.6%

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

       1. unpow1 99.6%

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

       2. fma-undefine 99.6%

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

       3. *-rgt-identity 99.6%

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

       4. distribute-lft-in 99.6%

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

       5. unpow2 99.6%

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

       6. fma-define 99.6%

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

   13. Simplified 99.6%

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

   if 1.00000000000000005e-4 < (*.f64 t t) < 4.99999999999999993e306

   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. Add Preprocessing

   3. Taylor expanded in x around 0 69.1%

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

      1. mul-1-neg 69.1%

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

      2. *-commutative 69.1%

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

      3. distribute-rgt-neg-in 69.1%

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

      4. *-commutative 69.1%

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

      5. distribute-rgt-neg-in 69.1%

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

   5. Simplified 69.1%

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

      1. associate-*r* 69.1%

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

      2. sqrt-prod 69.1%

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

      3. distribute-rgt-neg-out 69.1%

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

      4. sqrt-prod 69.1%

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

      5. add-sqr-sqrt 38.2%

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

      6. sqrt-unprod 39.7%

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

      7. sqr-neg 39.7%

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

      8. sqrt-unprod 8.8%

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

      9. add-sqr-sqrt 22.1%

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

      10. associate-*r* 22.1%

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

      11. *-commutative 22.1%

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

      12. *-commutative 22.1%

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

      13. associate-*l* 22.1%

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

      14. sqrt-prod 22.1%

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

      15. *-commutative 22.1%

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

      16. add-sqr-sqrt 8.8%

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

      17. sqrt-unprod 39.7%

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

      18. sqr-neg 39.7%

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

      19. sqrt-unprod 38.2%

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

      20. add-sqr-sqrt 69.1%

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

   7. Applied egg-rr 69.1%

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

      1. distribute-rgt-neg-in 69.1%

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

      2. *-commutative 69.1%

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

   9. Simplified 69.1%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 0.0001 \lor \neg \left(t \cdot t \leq 5 \cdot 10^{+306}\right):\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot \left(z \cdot \mathsf{fma}\left(t, t, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(-\sqrt{z \cdot 2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot x - y\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 0.00015:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t \leq 10^{+221}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(-t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* 0.5 x) y)) (t_2 (sqrt (* z 2.0))))
   (if (<= t 0.00015)
     (* t_1 t_2)
     (if (<= t 1e+221)
       (* (exp (/ (* t t) 2.0)) (* y (- t_2)))
       (* t_1 (* (* t (sqrt 2.0)) (sqrt z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double t_2 = sqrt((z * 2.0));
	double tmp;
	if (t <= 0.00015) {
		tmp = t_1 * t_2;
	} else if (t <= 1e+221) {
		tmp = exp(((t * t) / 2.0)) * (y * -t_2);
	} else {
		tmp = t_1 * ((t * sqrt(2.0)) * sqrt(z));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (0.5d0 * x) - y
    t_2 = sqrt((z * 2.0d0))
    if (t <= 0.00015d0) then
        tmp = t_1 * t_2
    else if (t <= 1d+221) then
        tmp = exp(((t * t) / 2.0d0)) * (y * -t_2)
    else
        tmp = t_1 * ((t * sqrt(2.0d0)) * sqrt(z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double t_2 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 0.00015) {
		tmp = t_1 * t_2;
	} else if (t <= 1e+221) {
		tmp = Math.exp(((t * t) / 2.0)) * (y * -t_2);
	} else {
		tmp = t_1 * ((t * Math.sqrt(2.0)) * Math.sqrt(z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (0.5 * x) - y
	t_2 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 0.00015:
		tmp = t_1 * t_2
	elif t <= 1e+221:
		tmp = math.exp(((t * t) / 2.0)) * (y * -t_2)
	else:
		tmp = t_1 * ((t * math.sqrt(2.0)) * math.sqrt(z))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(0.5 * x) - y)
	t_2 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 0.00015)
		tmp = Float64(t_1 * t_2);
	elseif (t <= 1e+221)
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(y * Float64(-t_2)));
	else
		tmp = Float64(t_1 * Float64(Float64(t * sqrt(2.0)) * sqrt(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (0.5 * x) - y;
	t_2 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 0.00015)
		tmp = t_1 * t_2;
	elseif (t <= 1e+221)
		tmp = exp(((t * t) / 2.0)) * (y * -t_2);
	else
		tmp = t_1 * ((t * sqrt(2.0)) * sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 0.00015], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t, 1e+221], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(y * (-t$95$2)), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.49999999999999987e-4

   1. Initial program 99.8%

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

      1. expm1-log1p-u 52.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\right)\right)} \]

      2. expm1-undefine 38.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\right)} - 1} \]

      3. associate-*l* 38.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)}\right)} - 1 \]

      4. exp-sqrt 38.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right)\right)} - 1 \]

      5. sqrt-unprod 38.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}}\right)} - 1 \]

      6. pow2 38.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{\color{blue}{{t}^{2}}}}\right)} - 1 \]

   4. Applied egg-rr 38.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{{t}^{2}}}\right)} - 1} \]
   5. Step-by-step derivation

      1. log1p-undefine 38.5%

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

      2. rem-exp-log 85.4%

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

      3. +-commutative 85.4%

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

      4. associate--l+ 99.8%

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

      5. metadata-eval 99.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{{t}^{2}}} + \color{blue}{0} \]

      6. +-rgt-identity 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l* 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 66.4%

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

   if 1.49999999999999987e-4 < t < 1e221

   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. Add Preprocessing

   3. Taylor expanded in x around 0 61.4%

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

      1. mul-1-neg 61.4%

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

      2. *-commutative 61.4%

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

      3. distribute-rgt-neg-in 61.4%

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

      4. *-commutative 61.4%

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

      5. distribute-rgt-neg-in 61.4%

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

   5. Simplified 61.4%

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

      1. associate-*r* 61.4%

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

      2. sqrt-prod 61.4%

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

      3. distribute-rgt-neg-out 61.4%

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

      4. sqrt-prod 61.4%

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

      5. add-sqr-sqrt 34.1%

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

      6. sqrt-unprod 41.1%

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

      7. sqr-neg 41.1%

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

      8. sqrt-unprod 11.4%

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

      9. add-sqr-sqrt 25.0%

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

      10. associate-*r* 25.0%

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

      11. *-commutative 25.0%

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

      12. *-commutative 25.0%

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

      13. associate-*l* 25.0%

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

      14. sqrt-prod 25.0%

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

      15. *-commutative 25.0%

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

      16. add-sqr-sqrt 11.4%

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

      17. sqrt-unprod 41.1%

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

      18. sqr-neg 41.1%

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

      19. sqrt-unprod 34.0%

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

      20. add-sqr-sqrt 61.4%

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

   7. Applied egg-rr 61.4%

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

      1. distribute-rgt-neg-in 61.4%

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

      2. *-commutative 61.4%

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

   9. Simplified 61.4%

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

   if 1e221 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 64.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\right)\right)} \]

      2. expm1-undefine 64.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\right)} - 1} \]

      3. associate-*l* 64.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)}\right)} - 1 \]

      4. exp-sqrt 64.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right)\right)} - 1 \]

      5. sqrt-unprod 64.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}}\right)} - 1 \]

      6. pow2 64.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{\color{blue}{{t}^{2}}}}\right)} - 1 \]

   4. Applied egg-rr 64.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{{t}^{2}}}\right)} - 1} \]
   5. Step-by-step derivation

      1. log1p-undefine 64.7%

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

      2. rem-exp-log 100.0%

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

      3. +-commutative 100.0%

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

      4. associate--l+ 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{{t}^{2}}} + \color{blue}{0} \]

      6. +-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 100.0%

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

      1. distribute-lft-out 100.0%

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

      2. *-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in t around inf 88.8%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.00015:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 10^{+221}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(-\sqrt{z \cdot 2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot x - y\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 11000:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+72}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(t\_1 \cdot \left(t \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* 0.5 x) y)) (t_2 (sqrt (* z 2.0))))
   (if (<= t 11000.0)
     (* t_1 t_2)
     (if (<= t 2.6e+72)
       (* (exp (/ (* t t) 2.0)) (* y t_2))
       (* (sqrt z) (* t_1 (* t (sqrt 2.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double t_2 = sqrt((z * 2.0));
	double tmp;
	if (t <= 11000.0) {
		tmp = t_1 * t_2;
	} else if (t <= 2.6e+72) {
		tmp = exp(((t * t) / 2.0)) * (y * t_2);
	} else {
		tmp = sqrt(z) * (t_1 * (t * sqrt(2.0)));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (0.5d0 * x) - y
    t_2 = sqrt((z * 2.0d0))
    if (t <= 11000.0d0) then
        tmp = t_1 * t_2
    else if (t <= 2.6d+72) then
        tmp = exp(((t * t) / 2.0d0)) * (y * t_2)
    else
        tmp = sqrt(z) * (t_1 * (t * sqrt(2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double t_2 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 11000.0) {
		tmp = t_1 * t_2;
	} else if (t <= 2.6e+72) {
		tmp = Math.exp(((t * t) / 2.0)) * (y * t_2);
	} else {
		tmp = Math.sqrt(z) * (t_1 * (t * Math.sqrt(2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (0.5 * x) - y
	t_2 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 11000.0:
		tmp = t_1 * t_2
	elif t <= 2.6e+72:
		tmp = math.exp(((t * t) / 2.0)) * (y * t_2)
	else:
		tmp = math.sqrt(z) * (t_1 * (t * math.sqrt(2.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(0.5 * x) - y)
	t_2 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 11000.0)
		tmp = Float64(t_1 * t_2);
	elseif (t <= 2.6e+72)
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(y * t_2));
	else
		tmp = Float64(sqrt(z) * Float64(t_1 * Float64(t * sqrt(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (0.5 * x) - y;
	t_2 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 11000.0)
		tmp = t_1 * t_2;
	elseif (t <= 2.6e+72)
		tmp = exp(((t * t) / 2.0)) * (y * t_2);
	else
		tmp = sqrt(z) * (t_1 * (t * sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 11000.0], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t, 2.6e+72], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(y * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[z], $MachinePrecision] * N[(t$95$1 * N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 11000

   1. Initial program 99.8%

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

      1. expm1-log1p-u 53.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\right)\right)} \]

      2. expm1-undefine 39.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\right)} - 1} \]

      3. associate-*l* 39.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)}\right)} - 1 \]

      4. exp-sqrt 39.2%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right)\right)} - 1 \]

      5. sqrt-unprod 39.2%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}}\right)} - 1 \]

      6. pow2 39.2%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{\color{blue}{{t}^{2}}}}\right)} - 1 \]

   4. Applied egg-rr 39.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{{t}^{2}}}\right)} - 1} \]
   5. Step-by-step derivation

      1. log1p-undefine 39.2%

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

      2. rem-exp-log 85.7%

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

      3. +-commutative 85.7%

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

      4. associate--l+ 99.8%

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

      5. metadata-eval 99.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{{t}^{2}}} + \color{blue}{0} \]

      6. +-rgt-identity 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l* 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 66.1%

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

   if 11000 < t < 2.59999999999999981e72

   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. Add Preprocessing

   3. Taylor expanded in x around 0 42.9%

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

      1. mul-1-neg 42.9%

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

      2. *-commutative 42.9%

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

      3. distribute-rgt-neg-in 42.9%

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

      4. *-commutative 42.9%

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

      5. distribute-rgt-neg-in 42.9%

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

   5. Simplified 42.9%

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

      1. pow1 42.9%

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

      2. *-commutative 42.9%

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

      3. *-commutative 42.9%

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

      4. associate-*l* 42.9%

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

      5. sqrt-prod 42.9%

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

      6. *-commutative 42.9%

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

      7. add-sqr-sqrt 14.3%

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

      8. sqrt-unprod 28.6%

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

      9. sqr-neg 28.6%

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

      10. sqrt-unprod 28.6%

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

      11. add-sqr-sqrt 42.9%

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

   7. Applied egg-rr 42.9%

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

      1. unpow1 42.9%

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

      2. *-commutative 42.9%

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

   9. Simplified 42.9%

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

   if 2.59999999999999981e72 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 52.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\right)\right)} \]

      2. expm1-undefine 52.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\right)} - 1} \]

      3. associate-*l* 52.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)}\right)} - 1 \]

      4. exp-sqrt 52.0%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right)\right)} - 1 \]

      5. sqrt-unprod 52.0%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}}\right)} - 1 \]

      6. pow2 52.0%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{\color{blue}{{t}^{2}}}}\right)} - 1 \]

   4. Applied egg-rr 52.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{{t}^{2}}}\right)} - 1} \]
   5. Step-by-step derivation

      1. log1p-undefine 52.0%

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

      2. rem-exp-log 100.0%

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

      3. +-commutative 100.0%

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

      4. associate--l+ 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{{t}^{2}}} + \color{blue}{0} \]

      6. +-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 62.0%

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

      1. distribute-lft-out 62.0%

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

      2. *-commutative 62.0%

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

   9. Simplified 62.0%

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

   10. Taylor expanded in t around inf 54.5%

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

       1. *-commutative 54.5%

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

       2. associate-*r* 54.5%

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

       3. *-commutative 54.5%

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

       4. *-commutative 54.5%

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

   12. Simplified 54.5%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 11000:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+72}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(0.5 \cdot x - y\right) \cdot \left(t \cdot \sqrt{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot x - y\\ \mathbf{if}\;t \leq 1:\\ \;\;\;\;t\_1 \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(t\_1 \cdot \left(t \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* 0.5 x) y)))
   (if (<= t 1.0)
     (* t_1 (sqrt (* z 2.0)))
     (* (sqrt z) (* t_1 (* t (sqrt 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double tmp;
	if (t <= 1.0) {
		tmp = t_1 * sqrt((z * 2.0));
	} else {
		tmp = sqrt(z) * (t_1 * (t * sqrt(2.0)));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (0.5d0 * x) - y
    if (t <= 1.0d0) then
        tmp = t_1 * sqrt((z * 2.0d0))
    else
        tmp = sqrt(z) * (t_1 * (t * sqrt(2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double tmp;
	if (t <= 1.0) {
		tmp = t_1 * Math.sqrt((z * 2.0));
	} else {
		tmp = Math.sqrt(z) * (t_1 * (t * Math.sqrt(2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (0.5 * x) - y
	tmp = 0
	if t <= 1.0:
		tmp = t_1 * math.sqrt((z * 2.0))
	else:
		tmp = math.sqrt(z) * (t_1 * (t * math.sqrt(2.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(0.5 * x) - y)
	tmp = 0.0
	if (t <= 1.0)
		tmp = Float64(t_1 * sqrt(Float64(z * 2.0)));
	else
		tmp = Float64(sqrt(z) * Float64(t_1 * Float64(t * sqrt(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (0.5 * x) - y;
	tmp = 0.0;
	if (t <= 1.0)
		tmp = t_1 * sqrt((z * 2.0));
	else
		tmp = sqrt(z) * (t_1 * (t * sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 1.0], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[z], $MachinePrecision] * N[(t$95$1 * N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1

   1. Initial program 99.8%

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

      1. expm1-log1p-u 53.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\right)\right)} \]

      2. expm1-undefine 38.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\right)} - 1} \]

      3. associate-*l* 38.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)}\right)} - 1 \]

      4. exp-sqrt 38.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right)\right)} - 1 \]

      5. sqrt-unprod 38.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}}\right)} - 1 \]

      6. pow2 38.9%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{\color{blue}{{t}^{2}}}}\right)} - 1 \]

   4. Applied egg-rr 38.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{{t}^{2}}}\right)} - 1} \]
   5. Step-by-step derivation

      1. log1p-undefine 38.9%

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

      2. rem-exp-log 85.6%

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

      3. +-commutative 85.6%

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

      4. associate--l+ 99.8%

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

      5. metadata-eval 99.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{{t}^{2}}} + \color{blue}{0} \]

      6. +-rgt-identity 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l* 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 66.4%

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

   if 1 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 51.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\right)\right)} \]

      2. expm1-undefine 51.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\right)} - 1} \]

      3. associate-*l* 51.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)}\right)} - 1 \]

      4. exp-sqrt 51.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right)\right)} - 1 \]

      5. sqrt-unprod 51.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}}\right)} - 1 \]

      6. pow2 51.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{\color{blue}{{t}^{2}}}}\right)} - 1 \]

   4. Applied egg-rr 51.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{{t}^{2}}}\right)} - 1} \]
   5. Step-by-step derivation

      1. log1p-undefine 51.7%

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

      2. rem-exp-log 100.0%

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

      3. +-commutative 100.0%

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

      4. associate--l+ 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{{t}^{2}}} + \color{blue}{0} \]

      6. +-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 54.0%

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

      1. distribute-lft-out 54.0%

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

      2. *-commutative 54.0%

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

   9. Simplified 54.0%

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

   10. Taylor expanded in t around inf 47.6%

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

       1. *-commutative 47.6%

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

       2. associate-*r* 47.6%

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

       3. *-commutative 47.6%

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

       4. *-commutative 47.6%

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

   12. Simplified 47.6%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(0.5 \cdot x - y\right) \cdot \left(t \cdot \sqrt{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return exp(((t * t) / 2.0)) * (((0.5 * x) - y) * sqrt((z * 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 = exp(((t * t) / 2.0d0)) * (((0.5d0 * x) - y) * sqrt((z * 2.0d0)))
end function

Java

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((z * 2.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 7: 58.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((0.5 * x) - y) * sqrt((z * 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 = ((0.5d0 * x) - y) * sqrt((z * 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. expm1-log1p-u 52.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\right)\right)} \]

   2. expm1-undefine 41.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\right)} - 1} \]

   3. associate-*l* 41.8%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)}\right)} - 1 \]

   4. exp-sqrt 41.8%

      \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right)\right)} - 1 \]

   5. sqrt-unprod 41.8%

      \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}}\right)} - 1 \]

   6. pow2 41.8%

      \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{\color{blue}{{t}^{2}}}}\right)} - 1 \]

4. Applied egg-rr 41.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{{t}^{2}}}\right)} - 1} \]
5. Step-by-step derivation

   1. log1p-undefine 41.8%

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

   2. rem-exp-log 88.9%

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

   3. +-commutative 88.9%

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

   4. associate--l+ 99.8%

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

   5. metadata-eval 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot e^{{t}^{2}}} + \color{blue}{0} \]

   6. +-rgt-identity 99.8%

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

   7. *-commutative 99.8%

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

   8. associate-*l* 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 54.2%

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

8. Final simplification 54.2%

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

Alternative 8: 31.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return y * -sqrt((z * 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 = y * -sqrt((z * 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 60.6%

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

   1. mul-1-neg 60.6%

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

   2. *-commutative 60.6%

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

   3. distribute-rgt-neg-in 60.6%

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

   4. *-commutative 60.6%

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

   5. distribute-rgt-neg-in 60.6%

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

5. Simplified 60.6%

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

6. Taylor expanded in t around 0 28.0%

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

   1. mul-1-neg 28.0%

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

   2. associate-*l* 28.0%

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

   3. *-commutative 28.0%

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

   4. distribute-rgt-neg-in 28.0%

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

   5. distribute-rgt-neg-in 28.0%

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

8. Simplified 28.0%

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

   1. pow1 28.0%

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

   2. distribute-rgt-neg-out 28.0%

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

   3. pow1/2 28.0%

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

   4. pow1/2 28.0%

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

   5. pow-prod-down 28.0%

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

10. Applied egg-rr 28.0%

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

    1. unpow1 28.0%

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

    2. unpow1/2 28.0%

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

    3. *-commutative 28.0%

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

12. Simplified 28.0%

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

13. Final simplification 28.0%

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

Developer target: 99.5% 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 2024079 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :alt
  (* (* (- (* 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))))