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

Percentage Accurate: 99.5% → 99.8%

Time: 18.6s

Alternatives: 13

Speedup: 1.0×

• 44.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.8%

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

   3. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. exp-sqrt 99.8%

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

   2. associate-*r* 99.4%

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

   3. *-commutative 99.4%

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

   4. expm1-log1p-u 55.6%

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

   5. expm1-udef 43.7%

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

6. Applied egg-rr 44.1%

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

   1. expm1-def 56.0%

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

   2. expm1-log1p 99.8%

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

   3. fma-neg 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*l* 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 89.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ t_2 := 0.5 \cdot x - y\\ \mathbf{if}\;t \cdot t \leq 5000000000000:\\ \;\;\;\;t\_2 \cdot \left(t\_1 \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{elif}\;t \cdot t \leq 10^{+146}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\left(0.5 \cdot x\right) \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \sqrt{2 \cdot \left(z \cdot {t}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))) (t_2 (- (* 0.5 x) y)))
   (if (<= (* t t) 5000000000000.0)
     (* t_2 (* t_1 (hypot 1.0 t)))
     (if (<= (* t t) 1e+146)
       (* (exp (/ (* t t) 2.0)) (* (* 0.5 x) t_1))
       (* t_2 (sqrt (* 2.0 (* z (pow t 2.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double t_2 = (0.5 * x) - y;
	double tmp;
	if ((t * t) <= 5000000000000.0) {
		tmp = t_2 * (t_1 * hypot(1.0, t));
	} else if ((t * t) <= 1e+146) {
		tmp = exp(((t * t) / 2.0)) * ((0.5 * x) * t_1);
	} else {
		tmp = t_2 * sqrt((2.0 * (z * pow(t, 2.0))));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((2.0 * z));
	double t_2 = (0.5 * x) - y;
	double tmp;
	if ((t * t) <= 5000000000000.0) {
		tmp = t_2 * (t_1 * Math.hypot(1.0, t));
	} else if ((t * t) <= 1e+146) {
		tmp = Math.exp(((t * t) / 2.0)) * ((0.5 * x) * t_1);
	} else {
		tmp = t_2 * Math.sqrt((2.0 * (z * Math.pow(t, 2.0))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	t_2 = (0.5 * x) - y
	tmp = 0
	if (t * t) <= 5000000000000.0:
		tmp = t_2 * (t_1 * math.hypot(1.0, t))
	elif (t * t) <= 1e+146:
		tmp = math.exp(((t * t) / 2.0)) * ((0.5 * x) * t_1)
	else:
		tmp = t_2 * math.sqrt((2.0 * (z * math.pow(t, 2.0))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	t_2 = Float64(Float64(0.5 * x) - y)
	tmp = 0.0
	if (Float64(t * t) <= 5000000000000.0)
		tmp = Float64(t_2 * Float64(t_1 * hypot(1.0, t)));
	elseif (Float64(t * t) <= 1e+146)
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(Float64(0.5 * x) * t_1));
	else
		tmp = Float64(t_2 * sqrt(Float64(2.0 * Float64(z * (t ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	t_2 = (0.5 * x) - y;
	tmp = 0.0;
	if ((t * t) <= 5000000000000.0)
		tmp = t_2 * (t_1 * hypot(1.0, t));
	elseif ((t * t) <= 1e+146)
		tmp = exp(((t * t) / 2.0)) * ((0.5 * x) * t_1);
	else
		tmp = t_2 * sqrt((2.0 * (z * (t ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 5000000000000.0], N[(t$95$2 * N[(t$95$1 * N[Sqrt[1.0 ^ 2 + t ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 1e+146], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[Sqrt[N[(2.0 * N[(z * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 5e12

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. exp-sqrt 99.6%

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

   3. Simplified 99.6%

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

      1. exp-sqrt 99.6%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

      4. expm1-log1p-u 58.8%

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

      5. expm1-udef 35.9%

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

   6. Applied egg-rr 35.9%

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

      1. expm1-def 58.8%

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

      2. expm1-log1p 99.6%

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

      3. fma-neg 99.6%

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

      4. *-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*l* 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in t around 0 99.5%

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

       1. distribute-rgt1-in 99.5%

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

       2. *-commutative 99.5%

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

       3. unpow2 99.5%

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

       4. fma-def 99.5%

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

   11. Simplified 99.5%

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

       1. associate-*r* 99.5%

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

       2. sqrt-prod 99.4%

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

   13. Applied egg-rr 99.4%

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

       1. *-commutative 99.4%

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

       2. fma-udef 99.4%

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

       3. unpow2 99.4%

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

       4. +-commutative 99.4%

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

       5. unpow2 99.4%

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

       6. hypot-1-def 99.4%

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

   15. Simplified 99.4%

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

   if 5e12 < (*.f64 t t) < 9.99999999999999934e145

   1. Initial program 95.2%

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

   3. Taylor expanded in x around inf 76.2%

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

      1. *-commutative 76.2%

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

      2. associate-*r* 76.2%

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

   5. Simplified 76.2%

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

      1. expm1-log1p-u 61.9%

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

      2. expm1-udef 38.1%

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

      3. *-commutative 38.1%

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

      4. associate-*l* 38.1%

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

      5. *-commutative 38.1%

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

      6. sqrt-unprod 38.1%

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

   7. Applied egg-rr 38.1%

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

      1. expm1-def 61.9%

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

      2. expm1-log1p 76.2%

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

      3. associate-*r* 76.2%

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

   9. Simplified 76.2%

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

   if 9.99999999999999934e145 < (*.f64 t 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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. exp-sqrt 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. expm1-log1p-u 50.0%

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

      5. expm1-udef 50.0%

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

   6. Applied egg-rr 50.0%

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

      1. expm1-def 50.0%

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

      2. expm1-log1p 100.0%

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

      3. fma-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 89.7%

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

       1. distribute-rgt1-in 89.7%

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

       2. *-commutative 89.7%

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

       3. unpow2 89.7%

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

       4. fma-def 89.7%

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

   11. Simplified 89.7%

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

   12. Taylor expanded in t around inf 89.7%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 5000000000000:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{elif}\;t \cdot t \leq 10^{+146}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\left(0.5 \cdot x\right) \cdot \sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot \left(z \cdot {t}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot x - y\\ \mathbf{if}\;t \cdot t \leq 5000000000000:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot \left(z \cdot \mathsf{fma}\left(t, t, 1\right)\right)}\\ \mathbf{elif}\;t \cdot t \leq 10^{+146}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\left(0.5 \cdot x\right) \cdot \sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot \left(z \cdot {t}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* 0.5 x) y)))
   (if (<= (* t t) 5000000000000.0)
     (* t_1 (sqrt (* 2.0 (* z (fma t t 1.0)))))
     (if (<= (* t t) 1e+146)
       (* (exp (/ (* t t) 2.0)) (* (* 0.5 x) (sqrt (* 2.0 z))))
       (* t_1 (sqrt (* 2.0 (* z (pow t 2.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double tmp;
	if ((t * t) <= 5000000000000.0) {
		tmp = t_1 * sqrt((2.0 * (z * fma(t, t, 1.0))));
	} else if ((t * t) <= 1e+146) {
		tmp = exp(((t * t) / 2.0)) * ((0.5 * x) * sqrt((2.0 * z)));
	} else {
		tmp = t_1 * sqrt((2.0 * (z * pow(t, 2.0))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(0.5 * x) - y)
	tmp = 0.0
	if (Float64(t * t) <= 5000000000000.0)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * Float64(z * fma(t, t, 1.0)))));
	elseif (Float64(t * t) <= 1e+146)
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(Float64(0.5 * x) * sqrt(Float64(2.0 * z))));
	else
		tmp = Float64(t_1 * sqrt(Float64(2.0 * Float64(z * (t ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 5000000000000.0], N[(t$95$1 * N[Sqrt[N[(2.0 * N[(z * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 1e+146], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * x), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(2.0 * N[(z * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 5e12

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. exp-sqrt 99.6%

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

   3. Simplified 99.6%

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

      1. exp-sqrt 99.6%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

      4. expm1-log1p-u 58.8%

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

      5. expm1-udef 35.9%

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

   6. Applied egg-rr 35.9%

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

      1. expm1-def 58.8%

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

      2. expm1-log1p 99.6%

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

      3. fma-neg 99.6%

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

      4. *-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*l* 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in t around 0 99.5%

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

       1. distribute-rgt1-in 99.5%

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

       2. *-commutative 99.5%

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

       3. unpow2 99.5%

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

       4. fma-def 99.5%

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

   11. Simplified 99.5%

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

   if 5e12 < (*.f64 t t) < 9.99999999999999934e145

   1. Initial program 95.2%

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

   3. Taylor expanded in x around inf 76.2%

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

      1. *-commutative 76.2%

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

      2. associate-*r* 76.2%

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

   5. Simplified 76.2%

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

      1. expm1-log1p-u 61.9%

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

      2. expm1-udef 38.1%

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

      3. *-commutative 38.1%

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

      4. associate-*l* 38.1%

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

      5. *-commutative 38.1%

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

      6. sqrt-unprod 38.1%

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

   7. Applied egg-rr 38.1%

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

      1. expm1-def 61.9%

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

      2. expm1-log1p 76.2%

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

      3. associate-*r* 76.2%

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

   9. Simplified 76.2%

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

   if 9.99999999999999934e145 < (*.f64 t 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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. exp-sqrt 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. expm1-log1p-u 50.0%

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

      5. expm1-udef 50.0%

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

   6. Applied egg-rr 50.0%

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

      1. expm1-def 50.0%

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

      2. expm1-log1p 100.0%

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

      3. fma-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 89.7%

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

       1. distribute-rgt1-in 89.7%

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

       2. *-commutative 89.7%

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

       3. unpow2 89.7%

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

       4. fma-def 89.7%

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

   11. Simplified 89.7%

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

   12. Taylor expanded in t around inf 89.7%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 5000000000000:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot \left(z \cdot \mathsf{fma}\left(t, t, 1\right)\right)}\\ \mathbf{elif}\;t \cdot t \leq 10^{+146}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\left(0.5 \cdot x\right) \cdot \sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot \left(z \cdot {t}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < 1

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 71.0%

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

   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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. exp-sqrt 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. expm1-log1p-u 58.9%

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

      5. expm1-udef 58.9%

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

   6. Applied egg-rr 58.9%

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

      1. expm1-def 58.9%

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

      2. expm1-log1p 100.0%

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

      3. fma-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 79.5%

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

       1. distribute-rgt1-in 79.5%

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

       2. *-commutative 79.5%

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

       3. unpow2 79.5%

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

       4. fma-def 79.5%

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

   11. Simplified 79.5%

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

   12. Taylor expanded in t around inf 79.5%

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

4. Final simplification 72.9%

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

Alternative 5: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot x - y\\ \mathbf{if}\;t \leq 3.2 \cdot 10^{+47}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{2 \cdot z} \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot \left(z \cdot {t}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* 0.5 x) y)))
   (if (<= t 3.2e+47)
     (* t_1 (* (sqrt (* 2.0 z)) (hypot 1.0 t)))
     (* t_1 (sqrt (* 2.0 (* z (pow t 2.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double tmp;
	if (t <= 3.2e+47) {
		tmp = t_1 * (sqrt((2.0 * z)) * hypot(1.0, t));
	} else {
		tmp = t_1 * sqrt((2.0 * (z * pow(t, 2.0))));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double tmp;
	if (t <= 3.2e+47) {
		tmp = t_1 * (Math.sqrt((2.0 * z)) * Math.hypot(1.0, t));
	} else {
		tmp = t_1 * Math.sqrt((2.0 * (z * Math.pow(t, 2.0))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (0.5 * x) - y
	tmp = 0
	if t <= 3.2e+47:
		tmp = t_1 * (math.sqrt((2.0 * z)) * math.hypot(1.0, t))
	else:
		tmp = t_1 * math.sqrt((2.0 * (z * math.pow(t, 2.0))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(0.5 * x) - y)
	tmp = 0.0
	if (t <= 3.2e+47)
		tmp = Float64(t_1 * Float64(sqrt(Float64(2.0 * z)) * hypot(1.0, t)));
	else
		tmp = Float64(t_1 * sqrt(Float64(2.0 * Float64(z * (t ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (0.5 * x) - y;
	tmp = 0.0;
	if (t <= 3.2e+47)
		tmp = t_1 * (sqrt((2.0 * z)) * hypot(1.0, t));
	else
		tmp = t_1 * sqrt((2.0 * (z * (t ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 3.2e+47], N[(t$95$1 * N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + t ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(2.0 * N[(z * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 3.2e47

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

      1. exp-sqrt 99.8%

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

      2. associate-*r* 99.3%

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

      3. *-commutative 99.3%

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

      4. expm1-log1p-u 55.0%

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

      5. expm1-udef 40.1%

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

   6. Applied egg-rr 40.6%

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

      1. expm1-def 55.5%

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

      2. expm1-log1p 99.8%

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

      3. fma-neg 99.8%

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

      4. *-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*l* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 90.7%

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

       1. distribute-rgt1-in 90.7%

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

       2. *-commutative 90.7%

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

       3. unpow2 90.7%

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

       4. fma-def 90.7%

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

   11. Simplified 90.7%

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

       1. associate-*r* 90.7%

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

       2. sqrt-prod 88.9%

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

   13. Applied egg-rr 88.9%

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

       1. *-commutative 88.9%

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

       2. fma-udef 88.9%

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

       3. unpow2 88.9%

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

       4. +-commutative 88.9%

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

       5. unpow2 88.9%

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

       6. hypot-1-def 82.0%

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

   15. Simplified 82.0%

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

   if 3.2e47 < 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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. exp-sqrt 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. expm1-log1p-u 57.7%

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

      5. expm1-udef 57.7%

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

   6. Applied egg-rr 57.7%

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

      1. expm1-def 57.7%

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

      2. expm1-log1p 100.0%

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

      3. fma-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 83.5%

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

       1. distribute-rgt1-in 83.5%

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

       2. *-commutative 83.5%

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

       3. unpow2 83.5%

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

       4. fma-def 83.5%

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

   11. Simplified 83.5%

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

   12. Taylor expanded in t around inf 83.5%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{+47}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot \left(z \cdot {t}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < 1

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 71.0%

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

   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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. exp-sqrt 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. expm1-log1p-u 58.9%

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

      5. expm1-udef 58.9%

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

   6. Applied egg-rr 58.9%

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

      1. expm1-def 58.9%

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

      2. expm1-log1p 100.0%

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

      3. fma-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 79.5%

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

       1. distribute-rgt1-in 79.5%

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

       2. *-commutative 79.5%

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

       3. unpow2 79.5%

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

       4. fma-def 79.5%

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

   11. Simplified 79.5%

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

   12. Taylor expanded in t around inf 52.3%

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

       1. *-commutative 52.3%

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

       2. fma-neg 52.3%

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

       3. associate-*r* 52.3%

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

       4. *-commutative 52.3%

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

       5. fma-neg 52.3%

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

   14. Simplified 52.3%

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

4. Final simplification 66.9%

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

Alternative 7: 59.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot x - y\\ \mathbf{if}\;t \leq 3.2:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* 0.5 x) y)))
   (if (<= t 3.2)
     (* t_1 (sqrt (* 2.0 z)))
     (* t_1 (cbrt (pow (* 2.0 z) 1.5))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double tmp;
	if (t <= 3.2) {
		tmp = t_1 * sqrt((2.0 * z));
	} else {
		tmp = t_1 * cbrt(pow((2.0 * z), 1.5));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double tmp;
	if (t <= 3.2) {
		tmp = t_1 * Math.sqrt((2.0 * z));
	} else {
		tmp = t_1 * Math.cbrt(Math.pow((2.0 * z), 1.5));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(0.5 * x) - y)
	tmp = 0.0
	if (t <= 3.2)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	else
		tmp = Float64(t_1 * cbrt((Float64(2.0 * z) ^ 1.5)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 3.2], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Power[N[Power[N[(2.0 * z), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 3.2000000000000002

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 71.0%

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

   if 3.2000000000000002 < 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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 18.1%

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

      1. *-commutative 18.1%

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

      2. *-commutative 18.1%

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

      3. associate-*l* 18.1%

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

   7. Simplified 18.1%

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

      1. add-cbrt-cube 24.5%

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

      2. pow1/3 24.5%

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

      3. *-commutative 24.5%

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

      4. sqrt-prod 24.5%

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

      5. *-commutative 24.5%

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

      6. sqrt-prod 24.5%

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

      7. *-commutative 24.5%

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

      8. sqrt-prod 24.5%

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

      9. add-sqr-sqrt 24.5%

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

      10. pow1 24.5%

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

      11. pow1/2 24.5%

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

      12. pow-prod-up 24.5%

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

      13. *-commutative 24.5%

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

      14. metadata-eval 24.5%

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

   9. Applied egg-rr 24.5%

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

       1. unpow1/3 24.5%

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

   11. Simplified 24.5%

       \[\leadsto \left(0.5 \cdot x - y\right) \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Final simplification 99.4%

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

Alternative 9: 57.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{+136}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{2} \cdot \left(0.5 \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.5e+136)
   (* (- (* 0.5 x) y) (sqrt (* 2.0 z)))
   (sqrt (* (pow x 2.0) (* 0.5 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.5e+136) {
		tmp = ((0.5 * x) - y) * sqrt((2.0 * z));
	} else {
		tmp = sqrt((pow(x, 2.0) * (0.5 * z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 1.5d+136) then
        tmp = ((0.5d0 * x) - y) * sqrt((2.0d0 * z))
    else
        tmp = sqrt(((x ** 2.0d0) * (0.5d0 * z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.5e+136) {
		tmp = ((0.5 * x) - y) * Math.sqrt((2.0 * z));
	} else {
		tmp = Math.sqrt((Math.pow(x, 2.0) * (0.5 * z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 1.5e+136:
		tmp = ((0.5 * x) - y) * math.sqrt((2.0 * z))
	else:
		tmp = math.sqrt((math.pow(x, 2.0) * (0.5 * z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.5e+136)
		tmp = Float64(Float64(Float64(0.5 * x) - y) * sqrt(Float64(2.0 * z)));
	else
		tmp = sqrt(Float64((x ^ 2.0) * Float64(0.5 * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 1.5e+136)
		tmp = ((0.5 * x) - y) * sqrt((2.0 * z));
	else
		tmp = sqrt(((x ^ 2.0) * (0.5 * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.5e+136], N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.5 * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.49999999999999989e136

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 67.3%

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

   if 1.49999999999999989e136 < 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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 10.2%

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

      1. sub-neg 10.2%

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

      2. *-commutative 10.2%

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

      3. fma-udef 10.2%

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

      4. add-sqr-sqrt 5.9%

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

      5. pow1 5.9%

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

      6. unpow2 5.9%

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

      7. metadata-eval 5.9%

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

      8. pow-prod-up 5.9%

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

      9. pow-prod-down 26.6%

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

   7. Applied egg-rr 35.0%

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

      1. unpow1/2 35.0%

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

      2. *-commutative 35.0%

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

   9. Simplified 35.0%

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

   10. Taylor expanded in x around inf 21.1%

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

       1. *-commutative 21.1%

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

       2. associate-*l* 21.1%

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

   12. Simplified 21.1%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{+136}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{2} \cdot \left(0.5 \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+158} \lor \neg \left(x \leq 2.8 \cdot 10^{-40}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (or (<= x -2.6e+158) (not (<= x 2.8e-40)))
     (* 0.5 (* x t_1))
     (* t_1 (- y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if ((x <= -2.6e+158) || !(x <= 2.8e-40)) {
		tmp = 0.5 * (x * t_1);
	} else {
		tmp = t_1 * -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * z))
    if ((x <= (-2.6d+158)) .or. (.not. (x <= 2.8d-40))) then
        tmp = 0.5d0 * (x * t_1)
    else
        tmp = t_1 * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((2.0 * z));
	double tmp;
	if ((x <= -2.6e+158) || !(x <= 2.8e-40)) {
		tmp = 0.5 * (x * t_1);
	} else {
		tmp = t_1 * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	tmp = 0
	if (x <= -2.6e+158) or not (x <= 2.8e-40):
		tmp = 0.5 * (x * t_1)
	else:
		tmp = t_1 * -y
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if ((x <= -2.6e+158) || !(x <= 2.8e-40))
		tmp = Float64(0.5 * Float64(x * t_1));
	else
		tmp = Float64(t_1 * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	tmp = 0.0;
	if ((x <= -2.6e+158) || ~((x <= 2.8e-40)))
		tmp = 0.5 * (x * t_1);
	else
		tmp = t_1 * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -2.6e+158], N[Not[LessEqual[x, 2.8e-40]], $MachinePrecision]], N[(0.5 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * (-y)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6e158 or 2.8e-40 < x

   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* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 67.5%

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

      1. sub-neg 67.5%

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

      2. distribute-lft-in 67.5%

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

      3. *-commutative 67.5%

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

      4. *-commutative 67.5%

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

      5. add-sqr-sqrt 28.9%

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

      6. sqrt-unprod 52.2%

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

      7. sqr-neg 52.2%

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

      8. sqrt-unprod 31.2%

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

      9. add-sqr-sqrt 51.2%

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

   7. Applied egg-rr 51.2%

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

      1. distribute-lft-out 52.2%

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

      2. fma-udef 52.2%

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

      3. *-commutative 52.2%

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

   9. Simplified 52.2%

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

   10. Taylor expanded in x around inf 53.6%

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

       1. associate-*r* 53.6%

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

       2. *-commutative 53.6%

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

   12. Simplified 53.6%

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

       1. expm1-log1p-u 35.2%

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

       2. expm1-udef 30.0%

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

       3. *-commutative 30.0%

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

       4. associate-*r* 30.0%

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

       5. associate-*l* 30.0%

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

       6. sqrt-prod 30.0%

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

   14. Applied egg-rr 30.0%

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

       1. expm1-def 35.3%

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

       2. expm1-log1p 53.7%

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

       3. associate-*l* 53.7%

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

       4. *-commutative 53.7%

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

   16. Simplified 53.7%

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

   if -2.6e158 < x < 2.8e-40

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 54.4%

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

   6. Taylor expanded in x around 0 42.6%

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

      1. mul-1-neg 42.6%

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

      2. associate-*l* 42.6%

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

      3. distribute-rgt-neg-in 42.6%

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

      4. *-commutative 42.6%

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

      5. distribute-rgt-neg-in 42.6%

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

   8. Simplified 42.6%

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

   10. Applied egg-rr 42.8%

       \[\leadsto \color{blue}{-y \cdot \sqrt{2 \cdot z}} \]
   11. Step-by-step derivation

       1. distribute-lft-neg-in 42.8%

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

   12. Simplified 42.8%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+158} \lor \neg \left(x \leq 2.8 \cdot 10^{-40}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{+144}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \left(y \cdot \left(x + y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 3.5e+144)
   (* (- (* 0.5 x) y) (sqrt (* 2.0 z)))
   (sqrt (* (* 2.0 z) (* y (+ x y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3.5e+144) {
		tmp = ((0.5 * x) - y) * sqrt((2.0 * z));
	} else {
		tmp = sqrt(((2.0 * z) * (y * (x + y))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 3.5d+144) then
        tmp = ((0.5d0 * x) - y) * sqrt((2.0d0 * z))
    else
        tmp = sqrt(((2.0d0 * z) * (y * (x + y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3.5e+144) {
		tmp = ((0.5 * x) - y) * Math.sqrt((2.0 * z));
	} else {
		tmp = Math.sqrt(((2.0 * z) * (y * (x + y))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 3.5e+144:
		tmp = ((0.5 * x) - y) * math.sqrt((2.0 * z))
	else:
		tmp = math.sqrt(((2.0 * z) * (y * (x + y))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 3.5e+144)
		tmp = Float64(Float64(Float64(0.5 * x) - y) * sqrt(Float64(2.0 * z)));
	else
		tmp = sqrt(Float64(Float64(2.0 * z) * Float64(y * Float64(x + y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 3.5e+144)
		tmp = ((0.5 * x) - y) * sqrt((2.0 * z));
	else
		tmp = sqrt(((2.0 * z) * (y * (x + y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 3.5e+144], N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(y * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.4999999999999998e144

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 67.3%

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

   if 3.4999999999999998e144 < 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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 10.2%

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

      1. sub-neg 10.2%

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

      2. *-commutative 10.2%

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

      3. fma-udef 10.2%

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

      4. add-sqr-sqrt 5.9%

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

      5. pow1 5.9%

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

      6. unpow2 5.9%

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

      7. metadata-eval 5.9%

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

      8. pow-prod-up 5.9%

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

      9. pow-prod-down 26.6%

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

   7. Applied egg-rr 35.0%

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

      1. unpow1/2 35.0%

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

      2. *-commutative 35.0%

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

   9. Simplified 35.0%

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

   10. Taylor expanded in x around 0 20.7%

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

       1. +-commutative 20.7%

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

       2. unpow2 20.7%

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

       3. distribute-rgt-out 20.7%

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

   12. Simplified 20.7%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{+144}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \left(y \cdot \left(x + y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 30.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.8%

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

   3. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around 0 59.5%

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

6. Taylor expanded in x around 0 33.1%

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

   1. mul-1-neg 33.1%

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

   2. associate-*l* 33.1%

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

   3. distribute-rgt-neg-in 33.1%

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

   4. *-commutative 33.1%

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

   5. distribute-rgt-neg-in 33.1%

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

8. Simplified 33.1%

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

10. Applied egg-rr 33.1%

    \[\leadsto \color{blue}{-y \cdot \sqrt{2 \cdot z}} \]
11. Step-by-step derivation

    1. distribute-lft-neg-in 33.1%

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

12. Simplified 33.1%

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

13. Final simplification 33.1%

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

Alternative 13: 2.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ y \cdot \sqrt{2 \cdot z} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.8%

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

   3. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around 0 59.5%

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

6. Taylor expanded in x around 0 33.1%

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

   1. mul-1-neg 33.1%

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

   2. associate-*l* 33.1%

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

   3. distribute-rgt-neg-in 33.1%

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

   4. *-commutative 33.1%

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

   5. distribute-rgt-neg-in 33.1%

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

8. Simplified 33.1%

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

10. Applied egg-rr 1.5%

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

    1. expm1-def 1.5%

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

    2. expm1-log1p 1.7%

       \[\leadsto \color{blue}{y \cdot \sqrt{2 \cdot z}} \]

12. Simplified 1.7%

    \[\leadsto \color{blue}{y \cdot \sqrt{2 \cdot z}} \]

13. Final simplification 1.7%

    \[\leadsto y \cdot \sqrt{2 \cdot z} \]
14. Add Preprocessing

Developer target: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024040 
(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))))