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

Percentage Accurate: 99.4% → 99.8%

Time: 17.3s

Alternatives: 11

Speedup: 1.0×

• 44.2% 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: 99.4% 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

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (* (- (* x 0.5) y) (sqrt (* (pow (exp t_m) t_m) (* 2.0 z)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[Power[N[Exp[t$95$m], $MachinePrecision], t$95$m], $MachinePrecision] * N[(2.0 * z), $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. Step-by-step derivation

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

   3. exp-prod 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.9%

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

   2. expm1-udef 74.9%

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

   3. sqrt-unprod 74.9%

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

   4. associate-*l* 74.9%

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

   5. pow-exp 74.9%

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

   6. pow2 74.9%

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

6. Applied egg-rr 74.9%

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

   1. expm1-def 98.9%

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

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-*l* 99.8%

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

8. Simplified 99.8%

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

   1. unpow2 99.8%

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

   2. exp-prod 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t\_m \leq 6.5 \cdot 10^{-31}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+74}:\\ \;\;\;\;t\_1 \cdot \sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(t\_m \cdot t\_2\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* 2.0 z))))
   (if (<= t_m 6.5e-31)
     (* t_1 t_2)
     (if (<= t_m 1.05e+74)
       (* t_1 (cbrt (pow (* 2.0 z) 1.5)))
       (* t_1 (* t_m t_2))))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt((2.0 * z));
	double tmp;
	if (t_m <= 6.5e-31) {
		tmp = t_1 * t_2;
	} else if (t_m <= 1.05e+74) {
		tmp = t_1 * cbrt(pow((2.0 * z), 1.5));
	} else {
		tmp = t_1 * (t_m * t_2);
	}
	return tmp;
}

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double t_2 = Math.sqrt((2.0 * z));
	double tmp;
	if (t_m <= 6.5e-31) {
		tmp = t_1 * t_2;
	} else if (t_m <= 1.05e+74) {
		tmp = t_1 * Math.cbrt(Math.pow((2.0 * z), 1.5));
	} else {
		tmp = t_1 * (t_m * t_2);
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (t_m <= 6.5e-31)
		tmp = Float64(t_1 * t_2);
	elseif (t_m <= 1.05e+74)
		tmp = Float64(t_1 * cbrt((Float64(2.0 * z) ^ 1.5)));
	else
		tmp = Float64(t_1 * Float64(t_m * t_2));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$m, 6.5e-31], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+74], N[(t$95$1 * N[Power[N[Power[N[(2.0 * z), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 6.49999999999999967e-31

   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. Step-by-step derivation

      1. associate-*l* 99.8%

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

      2. exp-sqrt 99.8%

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

      3. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 98.5%

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

      2. expm1-udef 65.6%

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

      3. sqrt-unprod 65.6%

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

      4. associate-*l* 65.6%

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

      5. pow-exp 65.6%

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

      6. pow2 65.6%

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

   6. Applied egg-rr 65.6%

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

      1. expm1-def 98.5%

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

      2. expm1-log1p 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 69.5%

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

   if 6.49999999999999967e-31 < t < 1.0499999999999999e74

   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. 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 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in t around 0 25.1%

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

      1. sqrt-prod 25.1%

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

      2. add-cbrt-cube 42.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt[3]{\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{z \cdot 2}}} \]

      3. pow1/3 41.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left(\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{z \cdot 2}\right)}^{0.3333333333333333}} \]

      4. add-sqr-sqrt 41.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left(\color{blue}{\left(z \cdot 2\right)} \cdot \sqrt{z \cdot 2}\right)}^{0.3333333333333333} \]

      5. pow1 41.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left(\color{blue}{{\left(z \cdot 2\right)}^{1}} \cdot \sqrt{z \cdot 2}\right)}^{0.3333333333333333} \]

      6. pow1/2 41.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left({\left(z \cdot 2\right)}^{1} \cdot \color{blue}{{\left(z \cdot 2\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      7. pow-prod-up 41.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\color{blue}{\left({\left(z \cdot 2\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      8. *-commutative 41.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left({\color{blue}{\left(2 \cdot z\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 41.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left({\left(2 \cdot z\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   7. Applied egg-rr 41.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left({\left(2 \cdot z\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 42.8%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}} \]

   9. Simplified 42.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}} \]

   if 1.0499999999999999e74 < 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. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. exp-sqrt 100.0%

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

      3. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow-exp 100.0%

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

      6. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 85.5%

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

       1. +-commutative 85.5%

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

   11. Simplified 85.5%

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

   12. Taylor expanded in t around inf 69.2%

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

       1. associate-*l* 55.2%

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

       2. *-commutative 55.2%

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

   14. Simplified 55.2%

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

       1. expm1-log1p-u 27.0%

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

       2. expm1-udef 27.0%

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

       3. *-commutative 27.0%

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

       4. associate-*r* 27.0%

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

       5. *-commutative 27.0%

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

       6. sqrt-prod 27.0%

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

       7. *-commutative 27.0%

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

       8. associate-*l* 28.6%

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

       9. fma-neg 28.6%

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

   16. Applied egg-rr 28.6%

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

       1. expm1-def 28.6%

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

       2. expm1-log1p 56.9%

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

       3. fma-neg 56.9%

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

       4. *-commutative 56.9%

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

       5. *-commutative 56.9%

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

       6. *-commutative 56.9%

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

   18. Simplified 56.9%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{-31}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+74}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(t \cdot \sqrt{2 \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 1.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t_m 1.0)
     (* t_1 (sqrt (* 2.0 z)))
     (* t_1 (sqrt (* (* 2.0 z) (pow t_m 2.0)))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$m, 1.0], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Power[t$95$m, 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. Step-by-step derivation

      1. associate-*l* 99.8%

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

      2. exp-sqrt 99.8%

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

      3. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 98.5%

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

      2. expm1-udef 65.3%

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

      3. sqrt-unprod 65.3%

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

      4. associate-*l* 65.3%

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

      5. pow-exp 65.2%

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

      6. pow2 65.2%

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

   6. Applied egg-rr 65.2%

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

      1. expm1-def 98.5%

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

      2. expm1-log1p 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 69.8%

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

   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. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. exp-sqrt 100.0%

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

      3. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow-exp 100.0%

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

      6. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 76.2%

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

       1. +-commutative 76.2%

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

   11. Simplified 76.2%

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

   12. Taylor expanded in t around inf 76.2%

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

       1. *-commutative 76.2%

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

       2. associate-*l* 76.2%

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

       3. *-commutative 76.2%

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

   14. Simplified 76.2%

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

4. Final simplification 71.6%

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

Alternative 4: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t\_m \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;t\_1 \cdot \left(\mathsf{hypot}\left(1, t\_m\right) \cdot \sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\left(2 \cdot z\right) \cdot {t\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t_m 1.4e+24)
     (* t_1 (* (hypot 1.0 t_m) (sqrt (* 2.0 z))))
     (* t_1 (sqrt (* (* 2.0 z) (pow t_m 2.0)))))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t_m <= 1.4e+24) {
		tmp = t_1 * (hypot(1.0, t_m) * sqrt((2.0 * z)));
	} else {
		tmp = t_1 * sqrt(((2.0 * z) * pow(t_m, 2.0)));
	}
	return tmp;
}

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t_m <= 1.4e+24) {
		tmp = t_1 * (Math.hypot(1.0, t_m) * Math.sqrt((2.0 * z)));
	} else {
		tmp = t_1 * Math.sqrt(((2.0 * z) * Math.pow(t_m, 2.0)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t_m <= 1.4e+24:
		tmp = t_1 * (math.hypot(1.0, t_m) * math.sqrt((2.0 * z)))
	else:
		tmp = t_1 * math.sqrt(((2.0 * z) * math.pow(t_m, 2.0)))
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t_m <= 1.4e+24)
		tmp = Float64(t_1 * Float64(hypot(1.0, t_m) * sqrt(Float64(2.0 * z))));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(2.0 * z) * (t_m ^ 2.0))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t_m <= 1.4e+24)
		tmp = t_1 * (hypot(1.0, t_m) * sqrt((2.0 * z)));
	else
		tmp = t_1 * sqrt(((2.0 * z) * (t_m ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$m, 1.4e+24], N[(t$95$1 * N[(N[Sqrt[1.0 ^ 2 + t$95$m ^ 2], $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.4000000000000001e24

   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. Step-by-step derivation

      1. associate-*l* 99.8%

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

      2. exp-sqrt 99.8%

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

      3. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 98.5%

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

      2. expm1-udef 66.2%

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

      3. sqrt-unprod 66.2%

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

      4. associate-*l* 66.2%

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

      5. pow-exp 66.2%

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

      6. pow2 66.2%

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

   6. Applied egg-rr 66.2%

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

      1. expm1-def 98.5%

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

      2. expm1-log1p 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 87.3%

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

       1. +-commutative 87.3%

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

   11. Simplified 87.3%

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

       1. sqrt-prod 86.8%

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

       2. +-commutative 86.8%

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

       3. unpow2 86.8%

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

       4. hypot-1-def 82.9%

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

   13. Applied egg-rr 82.9%

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

   if 1.4000000000000001e24 < 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. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. exp-sqrt 100.0%

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

      3. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow-exp 100.0%

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

      6. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 81.6%

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

       1. +-commutative 81.6%

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

   11. Simplified 81.6%

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

   12. Taylor expanded in t around inf 81.6%

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

       1. *-commutative 81.6%

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

       2. associate-*l* 81.6%

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

       3. *-commutative 81.6%

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

   14. Simplified 81.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\mathsf{hypot}\left(1, t\right) \cdot \sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot {t}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.1% accurate, 1.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t_m 1.0)
     (* t_1 (sqrt (* 2.0 z)))
     (* (* t_m (* t_1 (sqrt 2.0))) (sqrt z)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$m, 1.0], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$m * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $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. Step-by-step derivation

      1. associate-*l* 99.8%

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

      2. exp-sqrt 99.8%

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

      3. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 98.5%

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

      2. expm1-udef 65.3%

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

      3. sqrt-unprod 65.3%

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

      4. associate-*l* 65.3%

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

      5. pow-exp 65.2%

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

      6. pow2 65.2%

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

   6. Applied egg-rr 65.2%

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

      1. expm1-def 98.5%

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

      2. expm1-log1p 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 69.8%

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

   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. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. exp-sqrt 100.0%

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

      3. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow-exp 100.0%

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

      6. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 76.2%

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

       1. +-commutative 76.2%

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

   11. Simplified 76.2%

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

   12. Taylor expanded in t around inf 62.3%

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

4. Final simplification 67.8%

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

Alternative 6: 99.4% accurate, 1.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (* (exp (/ (* t_m t_m) 2.0)) (* (- (* x 0.5) y) (sqrt (* 2.0 z)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[Exp[N[(N[(t$95$m * t$95$m), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $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%

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

Alternative 7: 85.3% accurate, 1.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (* (- (* x 0.5) y) (sqrt (* (* 2.0 z) (fma t_m t_m 1.0)))))

C

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

Julia

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t$95$m * t$95$m + 1.0), $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. Step-by-step derivation

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

   3. exp-prod 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.9%

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

   2. expm1-udef 74.9%

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

   3. sqrt-unprod 74.9%

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

   4. associate-*l* 74.9%

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

   5. pow-exp 74.9%

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

   6. pow2 74.9%

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

6. Applied egg-rr 74.9%

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

   1. expm1-def 98.9%

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

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-*l* 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in t around 0 85.8%

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

    1. +-commutative 85.8%

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

    2. unpow2 85.8%

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

    3. fma-def 85.8%

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

11. Simplified 85.8%

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

12. Final simplification 85.8%

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

Alternative 8: 72.4% accurate, 1.9× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (* (- (* x 0.5) y) (sqrt (* 2.0 z)))))
   (if (<= t_m 1.0) t_1 (* t_m t_1))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$m, 1.0], t$95$1, N[(t$95$m * t$95$1), $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. Step-by-step derivation

      1. associate-*l* 99.8%

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

      2. exp-sqrt 99.8%

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

      3. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 98.5%

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

      2. expm1-udef 65.3%

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

      3. sqrt-unprod 65.3%

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

      4. associate-*l* 65.3%

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

      5. pow-exp 65.2%

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

      6. pow2 65.2%

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

   6. Applied egg-rr 65.2%

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

      1. expm1-def 98.5%

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

      2. expm1-log1p 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 69.8%

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

   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. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. exp-sqrt 100.0%

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

      3. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow-exp 100.0%

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

      6. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 76.2%

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

       1. +-commutative 76.2%

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

   11. Simplified 76.2%

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

   12. Taylor expanded in t around inf 62.3%

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

       1. associate-*l* 47.0%

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

       2. *-commutative 47.0%

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

   14. Simplified 47.0%

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

       1. expm1-log1p-u 23.1%

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

       2. expm1-udef 23.0%

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

       3. *-commutative 23.0%

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

       4. associate-*r* 23.0%

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

       5. *-commutative 23.0%

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

       6. sqrt-prod 23.0%

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

       7. *-commutative 23.0%

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

       8. associate-*l* 24.3%

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

       9. fma-neg 24.3%

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

   16. Applied egg-rr 24.3%

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

       1. expm1-def 24.4%

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

       2. expm1-log1p 48.4%

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

       3. associate-*r* 47.0%

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

       4. *-commutative 47.0%

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

       5. fma-neg 47.0%

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

       6. *-commutative 47.0%

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

       7. *-commutative 47.0%

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

   18. Simplified 47.0%

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

4. Final simplification 63.5%

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

Alternative 9: 73.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t\_m \leq 1:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(t\_m \cdot t\_2\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* 2.0 z))))
   (if (<= t_m 1.0) (* t_1 t_2) (* t_1 (* t_m t_2)))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt((2.0 * z));
	double tmp;
	if (t_m <= 1.0) {
		tmp = t_1 * t_2;
	} else {
		tmp = t_1 * (t_m * t_2);
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    t_2 = sqrt((2.0d0 * z))
    if (t_m <= 1.0d0) then
        tmp = t_1 * t_2
    else
        tmp = t_1 * (t_m * t_2)
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double t_2 = Math.sqrt((2.0 * z));
	double tmp;
	if (t_m <= 1.0) {
		tmp = t_1 * t_2;
	} else {
		tmp = t_1 * (t_m * t_2);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = (x * 0.5) - y
	t_2 = math.sqrt((2.0 * z))
	tmp = 0
	if t_m <= 1.0:
		tmp = t_1 * t_2
	else:
		tmp = t_1 * (t_m * t_2)
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (t_m <= 1.0)
		tmp = Float64(t_1 * t_2);
	else
		tmp = Float64(t_1 * Float64(t_m * t_2));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = (x * 0.5) - y;
	t_2 = sqrt((2.0 * z));
	tmp = 0.0;
	if (t_m <= 1.0)
		tmp = t_1 * t_2;
	else
		tmp = t_1 * (t_m * t_2);
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$m, 1.0], N[(t$95$1 * t$95$2), $MachinePrecision], N[(t$95$1 * N[(t$95$m * t$95$2), $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. Step-by-step derivation

      1. associate-*l* 99.8%

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

      2. exp-sqrt 99.8%

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

      3. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 98.5%

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

      2. expm1-udef 65.3%

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

      3. sqrt-unprod 65.3%

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

      4. associate-*l* 65.3%

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

      5. pow-exp 65.2%

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

      6. pow2 65.2%

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

   6. Applied egg-rr 65.2%

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

      1. expm1-def 98.5%

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

      2. expm1-log1p 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 69.8%

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

   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. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. exp-sqrt 100.0%

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

      3. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow-exp 100.0%

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

      6. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 76.2%

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

       1. +-commutative 76.2%

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

   11. Simplified 76.2%

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

   12. Taylor expanded in t around inf 62.3%

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

       1. associate-*l* 47.0%

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

       2. *-commutative 47.0%

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

   14. Simplified 47.0%

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

       1. expm1-log1p-u 23.1%

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

       2. expm1-udef 23.0%

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

       3. *-commutative 23.0%

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

       4. associate-*r* 23.0%

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

       5. *-commutative 23.0%

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

       6. sqrt-prod 23.0%

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

       7. *-commutative 23.0%

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

       8. associate-*l* 24.3%

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

       9. fma-neg 24.3%

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

   16. Applied egg-rr 24.3%

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

       1. expm1-def 24.4%

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

       2. expm1-log1p 48.4%

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

       3. fma-neg 48.4%

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

       4. *-commutative 48.4%

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

       5. *-commutative 48.4%

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

       6. *-commutative 48.4%

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

   18. Simplified 48.4%

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

4. Final simplification 63.9%

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

Alternative 10: 57.3% accurate, 2.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m) :precision binary64 (* (- (* x 0.5) y) (sqrt (* 2.0 z))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $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. Step-by-step derivation

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

   3. exp-prod 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.9%

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

   2. expm1-udef 74.9%

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

   3. sqrt-unprod 74.9%

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

   4. associate-*l* 74.9%

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

   5. pow-exp 74.9%

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

   6. pow2 74.9%

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

6. Applied egg-rr 74.9%

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

   1. expm1-def 98.9%

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

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-*l* 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in t around 0 54.3%

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

10. Final simplification 54.3%

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

Alternative 11: 30.2% accurate, 2.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m) :precision binary64 (* 0.5 (* x (sqrt (* 2.0 z)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(0.5 * N[(x * N[Sqrt[N[(2.0 * z), $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. Step-by-step derivation

   1. *-commutative 99.8%

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

   2. associate-*l* 99.5%

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

   3. exp-sqrt 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in t around 0 54.2%

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

   1. associate-*r* 54.1%

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

   2. *-commutative 54.1%

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

   3. *-commutative 54.1%

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

   4. fma-neg 54.1%

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

   5. associate-*l* 54.1%

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

   6. fma-neg 54.1%

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

   7. *-commutative 54.1%

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

7. Simplified 54.1%

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

8. Taylor expanded in x around inf 29.2%

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

   1. expm1-log1p-u 19.6%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \]

   2. expm1-udef 14.1%

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

   3. associate-*l* 14.1%

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

   4. sqrt-prod 14.1%

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

10. Applied egg-rr 14.1%

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

    1. expm1-def 19.7%

       \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \sqrt{2 \cdot z}\right)\right)} \]

    2. expm1-log1p 29.2%

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

12. Simplified 29.2%

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

13. Final simplification 29.2%

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

Developer target: 99.4% 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 2024031 
(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))))