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

Percentage Accurate: 99.4% → 99.4%

Time: 38.6s

Alternatives: 18

Speedup: 1.1×

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

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 * (-0.5d0)))
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(-t), (t * -0.5));
}

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * (exp(-t) ^ (t * -0.5));
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[(-t)], $MachinePrecision], N[(t * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. div-inv N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft-neg-in N/A

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

   7. associate-*l* N/A

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

   8. exp-prod N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

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) ** (0.5d0 * t))
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(t), (0.5 * t));
}

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * (exp(t) ^ (0.5 * t));
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[t], $MachinePrecision], N[(0.5 * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. exp-prod N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 97.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \cdot t \leq 2 \cdot 10^{-12}:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot \mathsf{fma}\left(t, z \cdot t, z\right)}\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot e^{t \cdot t}\right)} \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(t\_1 \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right), 0.5\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* t t) 2e-12)
     (* t_1 (sqrt (* 2.0 (fma t (* z t) z))))
     (if (<= (* t t) 5e+85)
       (* (sqrt (* 2.0 (* z (exp (* t t))))) (* x 0.5))
       (*
        (sqrt (* z 2.0))
        (*
         t_1
         (fma
          (* t t)
          (fma (* t t) (fma (* t t) 0.020833333333333332 0.125) 0.5)
          1.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((t * t) <= 2e-12) {
		tmp = t_1 * sqrt((2.0 * fma(t, (z * t), z)));
	} else if ((t * t) <= 5e+85) {
		tmp = sqrt((2.0 * (z * exp((t * t))))) * (x * 0.5);
	} else {
		tmp = sqrt((z * 2.0)) * (t_1 * fma((t * t), fma((t * t), fma((t * t), 0.020833333333333332, 0.125), 0.5), 1.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(t * t) <= 2e-12)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * fma(t, Float64(z * t), z))));
	elseif (Float64(t * t) <= 5e+85)
		tmp = Float64(sqrt(Float64(2.0 * Float64(z * exp(Float64(t * t))))) * Float64(x * 0.5));
	else
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(t_1 * fma(Float64(t * t), fma(Float64(t * t), fma(Float64(t * t), 0.020833333333333332, 0.125), 0.5), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 2e-12], N[(t$95$1 * N[Sqrt[N[(2.0 * N[(t * N[(z * t), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 5e+85], N[(N[Sqrt[N[(2.0 * N[(z * N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.020833333333333332 + 0.125), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 1.99999999999999996e-12

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. exp-sqrt N/A

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

      9. sqrt-unprod N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if 1.99999999999999996e-12 < (*.f64 t t) < 5.0000000000000001e85

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. exp-sqrt N/A

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

      9. sqrt-unprod N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 95.8

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 95.8

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

   7. Applied rewrites 95.8%

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

   if 5.0000000000000001e85 < (*.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. associate-*l* N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)}\right) \cdot \sqrt{z \cdot 2} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\left({t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) + 1\right)}\right) \cdot \sqrt{z \cdot 2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\mathsf{fma}\left({t}^{2}, \frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right), 1\right)}\right) \cdot \sqrt{z \cdot 2} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right), 1\right)\right) \cdot \sqrt{z \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right), 1\right)\right) \cdot \sqrt{z \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(t \cdot t, \color{blue}{{t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) + \frac{1}{2}}, 1\right)\right) \cdot \sqrt{z \cdot 2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(t \cdot t, \color{blue}{\mathsf{fma}\left({t}^{2}, \frac{1}{8} + \frac{1}{48} \cdot {t}^{2}, \frac{1}{2}\right)}, 1\right)\right) \cdot \sqrt{z \cdot 2} \]

      7. unpow2 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{8} + \frac{1}{48} \cdot {t}^{2}, \frac{1}{2}\right), 1\right)\right) \cdot \sqrt{z \cdot 2} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{8} + \frac{1}{48} \cdot {t}^{2}, \frac{1}{2}\right), 1\right)\right) \cdot \sqrt{z \cdot 2} \]

      9. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \color{blue}{\frac{1}{48} \cdot {t}^{2} + \frac{1}{8}}, \frac{1}{2}\right), 1\right)\right) \cdot \sqrt{z \cdot 2} \]

      10. *-commutative N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \color{blue}{{t}^{2} \cdot \frac{1}{48}} + \frac{1}{8}, \frac{1}{2}\right), 1\right)\right) \cdot \sqrt{z \cdot 2} \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \color{blue}{\mathsf{fma}\left({t}^{2}, \frac{1}{48}, \frac{1}{8}\right)}, \frac{1}{2}\right), 1\right)\right) \cdot \sqrt{z \cdot 2} \]

      12. unpow2 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{48}, \frac{1}{8}\right), \frac{1}{2}\right), 1\right)\right) \cdot \sqrt{z \cdot 2} \]

      13. lower-*.f64 100.0

          \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, 0.020833333333333332, 0.125\right), 0.5\right), 1\right)\right) \cdot \sqrt{z \cdot 2} \]

   9. Applied rewrites 100.0%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right), 0.5\right), 1\right)}\right) \cdot \sqrt{z \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(t, z \cdot t, z\right)}\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot e^{t \cdot t}\right)} \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right), 0.5\right), 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 99.8

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 5: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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((2.0d0 * (z * exp((t * t)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. exp-sqrt N/A

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

   9. sqrt-unprod N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-exp.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 6: 95.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \sqrt{z \cdot 2} \cdot \left(\mathsf{fma}\left(0.5, x, \left(t \cdot t\right) \cdot \mathsf{fma}\left(0.5, t\_1, t\_1 \cdot \left(t \cdot \left(t \cdot \mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right)\right)\right)\right)\right) - y\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (*
    (sqrt (* z 2.0))
    (-
     (fma
      0.5
      x
      (*
       (* t t)
       (fma
        0.5
        t_1
        (* t_1 (* t (* t (fma (* t t) 0.020833333333333332 0.125)))))))
     y))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	return sqrt((z * 2.0)) * (fma(0.5, x, ((t * t) * fma(0.5, t_1, (t_1 * (t * (t * fma((t * t), 0.020833333333333332, 0.125))))))) - y);
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	return Float64(sqrt(Float64(z * 2.0)) * Float64(fma(0.5, x, Float64(Float64(t * t) * fma(0.5, t_1, Float64(t_1 * Float64(t * Float64(t * fma(Float64(t * t), 0.020833333333333332, 0.125))))))) - y))
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * x + N[(N[(t * t), $MachinePrecision] * N[(0.5 * t$95$1 + N[(t$95$1 * N[(t * N[(t * N[(N[(t * t), $MachinePrecision] * 0.020833333333333332 + 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. div-inv N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft-neg-in N/A

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

   7. associate-*l* N/A

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

   8. exp-prod N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot x + {t}^{2} \cdot \left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot x - y\right) + {t}^{2} \cdot \left(\frac{1}{48} \cdot \left({t}^{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \frac{1}{8} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)\right) - y\right)} \cdot \sqrt{z \cdot 2} \]
8. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot x + {t}^{2} \cdot \left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot x - y\right) + {t}^{2} \cdot \left(\frac{1}{48} \cdot \left({t}^{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \frac{1}{8} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)\right) - y\right)} \cdot \sqrt{z \cdot 2} \]

9. Applied rewrites 95.5%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, x, \left(t \cdot t\right) \cdot \mathsf{fma}\left(0.5, 0.5 \cdot x - y, \left(0.5 \cdot x - y\right) \cdot \left(t \cdot \left(t \cdot \mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right)\right)\right)\right)\right) - y\right)} \cdot \sqrt{z \cdot 2} \]

10. Final simplification 95.5%

    \[\leadsto \sqrt{z \cdot 2} \cdot \left(\mathsf{fma}\left(0.5, x, \left(t \cdot t\right) \cdot \mathsf{fma}\left(0.5, x \cdot 0.5 - y, \left(x \cdot 0.5 - y\right) \cdot \left(t \cdot \left(t \cdot \mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right)\right)\right)\right)\right) - y\right) \]
11. Add Preprocessing

Alternative 7: 95.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{z \cdot 2} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right), 0.5\right), 1\right)\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (sqrt (* z 2.0))
  (*
   (- (* x 0.5) y)
   (fma
    (* t t)
    (fma (* t t) (fma (* t t) 0.020833333333333332 0.125) 0.5)
    1.0))))

C

double code(double x, double y, double z, double t) {
	return sqrt((z * 2.0)) * (((x * 0.5) - y) * fma((t * t), fma((t * t), fma((t * t), 0.020833333333333332, 0.125), 0.5), 1.0));
}

Julia

function code(x, y, z, t)
	return Float64(sqrt(Float64(z * 2.0)) * Float64(Float64(Float64(x * 0.5) - y) * fma(Float64(t * t), fma(Float64(t * t), fma(Float64(t * t), 0.020833333333333332, 0.125), 0.5), 1.0)))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.020833333333333332 + 0.125), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. div-inv N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft-neg-in N/A

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

   7. associate-*l* N/A

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

   8. exp-prod N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)}\right) \cdot \sqrt{z \cdot 2} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\left({t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) + 1\right)}\right) \cdot \sqrt{z \cdot 2} \]

   2. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\mathsf{fma}\left({t}^{2}, \frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right), 1\right)}\right) \cdot \sqrt{z \cdot 2} \]

   3. unpow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right), 1\right)\right) \cdot \sqrt{z \cdot 2} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right), 1\right)\right) \cdot \sqrt{z \cdot 2} \]

   5. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(t \cdot t, \color{blue}{{t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) + \frac{1}{2}}, 1\right)\right) \cdot \sqrt{z \cdot 2} \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(t \cdot t, \color{blue}{\mathsf{fma}\left({t}^{2}, \frac{1}{8} + \frac{1}{48} \cdot {t}^{2}, \frac{1}{2}\right)}, 1\right)\right) \cdot \sqrt{z \cdot 2} \]

   7. unpow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{8} + \frac{1}{48} \cdot {t}^{2}, \frac{1}{2}\right), 1\right)\right) \cdot \sqrt{z \cdot 2} \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{8} + \frac{1}{48} \cdot {t}^{2}, \frac{1}{2}\right), 1\right)\right) \cdot \sqrt{z \cdot 2} \]

   9. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \color{blue}{\frac{1}{48} \cdot {t}^{2} + \frac{1}{8}}, \frac{1}{2}\right), 1\right)\right) \cdot \sqrt{z \cdot 2} \]

   10. *-commutative N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \color{blue}{{t}^{2} \cdot \frac{1}{48}} + \frac{1}{8}, \frac{1}{2}\right), 1\right)\right) \cdot \sqrt{z \cdot 2} \]

   11. lower-fma.f64 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \color{blue}{\mathsf{fma}\left({t}^{2}, \frac{1}{48}, \frac{1}{8}\right)}, \frac{1}{2}\right), 1\right)\right) \cdot \sqrt{z \cdot 2} \]

   12. unpow2 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{48}, \frac{1}{8}\right), \frac{1}{2}\right), 1\right)\right) \cdot \sqrt{z \cdot 2} \]

   13. lower-*.f64 95.5

       \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, 0.020833333333333332, 0.125\right), 0.5\right), 1\right)\right) \cdot \sqrt{z \cdot 2} \]

9. Applied rewrites 95.5%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right), 0.5\right), 1\right)}\right) \cdot \sqrt{z \cdot 2} \]

10. Final simplification 95.5%

    \[\leadsto \sqrt{z \cdot 2} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right), 0.5\right), 1\right)\right) \]
11. Add Preprocessing

Alternative 8: 93.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x \cdot 0.5 - y\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t, t \cdot 0.125, 0.5\right), t \cdot t, 1\right)\right) \cdot \sqrt{z}\right) \cdot \sqrt{2} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (* (- (* x 0.5) y) (fma (fma t (* t 0.125) 0.5) (* t t) 1.0)) (sqrt z))
  (sqrt 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((((x * 0.5) - y) * fma(fma(t, (t * 0.125), 0.5), (t * t), 1.0)) * sqrt(z)) * sqrt(2.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(Float64(x * 0.5) - y) * fma(fma(t, Float64(t * 0.125), 0.5), Float64(t * t), 1.0)) * sqrt(z)) * sqrt(2.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[(N[(t * N[(t * 0.125), $MachinePrecision] + 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left({t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right) + 1\right)} \]

   2. unpow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right) + 1\right) \]

   3. associate-*l* N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\color{blue}{t \cdot \left(t \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right)\right)} + 1\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t, t \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right), 1\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, \color{blue}{t \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right)}, 1\right) \]

   6. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \color{blue}{\left(\frac{1}{8} \cdot {t}^{2} + \frac{1}{2}\right)}, 1\right) \]

   7. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \left(\color{blue}{{t}^{2} \cdot \frac{1}{8}} + \frac{1}{2}\right), 1\right) \]

   8. unpow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{1}{8} + \frac{1}{2}\right), 1\right) \]

   9. associate-*l* N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \left(\color{blue}{t \cdot \left(t \cdot \frac{1}{8}\right)} + \frac{1}{2}\right), 1\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \color{blue}{\mathsf{fma}\left(t, t \cdot \frac{1}{8}, \frac{1}{2}\right)}, 1\right) \]

   11. lower-*.f64 93.3

       \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, \color{blue}{t \cdot 0.125}, 0.5\right), 1\right) \]

5. Applied rewrites 93.3%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot 0.125, 0.5\right), 1\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot \frac{1}{8}, \frac{1}{2}\right), 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot \frac{1}{8}, \frac{1}{2}\right), 1\right) \cdot \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot \frac{1}{8}, \frac{1}{2}\right), 1\right) \cdot \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot \frac{1}{8}, \frac{1}{2}\right), 1\right) \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot \sqrt{z \cdot 2}} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot \frac{1}{8}, \frac{1}{2}\right), 1\right) \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot \color{blue}{\sqrt{z \cdot 2}} \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot \frac{1}{8}, \frac{1}{2}\right), 1\right) \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot \sqrt{\color{blue}{z \cdot 2}} \]

   7. sqrt-prod N/A

      \[\leadsto \left(\mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot \frac{1}{8}, \frac{1}{2}\right), 1\right) \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot \frac{1}{8}, \frac{1}{2}\right), 1\right) \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot \left(\color{blue}{\sqrt{z}} \cdot \sqrt{2}\right) \]

   9. lift-sqrt.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot \frac{1}{8}, \frac{1}{2}\right), 1\right) \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot \left(\sqrt{z} \cdot \color{blue}{\sqrt{2}}\right) \]

   10. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot \frac{1}{8}, \frac{1}{2}\right), 1\right) \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot \sqrt{z}\right) \cdot \sqrt{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot \frac{1}{8}, \frac{1}{2}\right), 1\right) \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot \sqrt{z}\right) \cdot \sqrt{2}} \]

7. Applied rewrites 93.8%

   \[\leadsto \color{blue}{\left(\left(\left(x \cdot 0.5 - y\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t, t \cdot 0.125, 0.5\right), t \cdot t, 1\right)\right) \cdot \sqrt{z}\right) \cdot \sqrt{2}} \]
8. Add Preprocessing

Alternative 9: 92.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot 0.125, 0.5\right), 1\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (- (* x 0.5) y) (sqrt (* z 2.0)))
  (fma t (* t (fma t (* t 0.125) 0.5)) 1.0)))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * fma(t, (t * fma(t, (t * 0.125), 0.5)), 1.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * fma(t, Float64(t * fma(t, Float64(t * 0.125), 0.5)), 1.0))
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[(t * N[(t * N[(t * N[(t * 0.125), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left({t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right) + 1\right)} \]

   2. unpow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right) + 1\right) \]

   3. associate-*l* N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\color{blue}{t \cdot \left(t \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right)\right)} + 1\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t, t \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right), 1\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, \color{blue}{t \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right)}, 1\right) \]

   6. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \color{blue}{\left(\frac{1}{8} \cdot {t}^{2} + \frac{1}{2}\right)}, 1\right) \]

   7. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \left(\color{blue}{{t}^{2} \cdot \frac{1}{8}} + \frac{1}{2}\right), 1\right) \]

   8. unpow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{1}{8} + \frac{1}{2}\right), 1\right) \]

   9. associate-*l* N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \left(\color{blue}{t \cdot \left(t \cdot \frac{1}{8}\right)} + \frac{1}{2}\right), 1\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \color{blue}{\mathsf{fma}\left(t, t \cdot \frac{1}{8}, \frac{1}{2}\right)}, 1\right) \]

   11. lower-*.f64 93.3

       \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, \color{blue}{t \cdot 0.125}, 0.5\right), 1\right) \]

5. Applied rewrites 93.3%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot 0.125, 0.5\right), 1\right)} \]
6. Add Preprocessing

Alternative 10: 91.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \left(\left(t \cdot t\right) \cdot 0.125\right), 1\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (fma t (* t (* (* t t) 0.125)) 1.0)))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * fma(t, (t * ((t * t) * 0.125)), 1.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * fma(t, Float64(t * Float64(Float64(t * t) * 0.125)), 1.0))
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[(t * N[(t * N[(N[(t * t), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left({t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right) + 1\right)} \]

   2. unpow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right) + 1\right) \]

   3. associate-*l* N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\color{blue}{t \cdot \left(t \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right)\right)} + 1\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t, t \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right), 1\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, \color{blue}{t \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right)}, 1\right) \]

   6. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \color{blue}{\left(\frac{1}{8} \cdot {t}^{2} + \frac{1}{2}\right)}, 1\right) \]

   7. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \left(\color{blue}{{t}^{2} \cdot \frac{1}{8}} + \frac{1}{2}\right), 1\right) \]

   8. unpow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{1}{8} + \frac{1}{2}\right), 1\right) \]

   9. associate-*l* N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \left(\color{blue}{t \cdot \left(t \cdot \frac{1}{8}\right)} + \frac{1}{2}\right), 1\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \color{blue}{\mathsf{fma}\left(t, t \cdot \frac{1}{8}, \frac{1}{2}\right)}, 1\right) \]

   11. lower-*.f64 93.3

       \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, \color{blue}{t \cdot 0.125}, 0.5\right), 1\right) \]

5. Applied rewrites 93.3%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot 0.125, 0.5\right), 1\right)} \]

6. Taylor expanded in t around inf

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, \frac{1}{8} \cdot \color{blue}{{t}^{3}}, 1\right) \]
7. Step-by-step derivation

8. Applied rewrites 93.1%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot 0.125\right)}, 1\right) \]
9. Add Preprocessing

Alternative 11: 65.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot \left(z \cdot \mathsf{fma}\left(t, t, 1\right)\right)}\\ \mathbf{if}\;t \leq 1.7 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{elif}\;t \leq 10^{+227}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 (* z (fma t t 1.0))))))
   (if (<= t 1.7e+30)
     (* (sqrt (* z 2.0)) (- (* x 0.5) y))
     (if (<= t 1e+227) (* (* x 0.5) t_1) (* t_1 (- y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * (z * fma(t, t, 1.0))));
	double tmp;
	if (t <= 1.7e+30) {
		tmp = sqrt((z * 2.0)) * ((x * 0.5) - y);
	} else if (t <= 1e+227) {
		tmp = (x * 0.5) * t_1;
	} else {
		tmp = t_1 * -y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * Float64(z * fma(t, t, 1.0))))
	tmp = 0.0
	if (t <= 1.7e+30)
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(Float64(x * 0.5) - y));
	elseif (t <= 1e+227)
		tmp = Float64(Float64(x * 0.5) * t_1);
	else
		tmp = Float64(t_1 * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(z * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1.7e+30], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e+227], N[(N[(x * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * (-y)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.7000000000000001e30

   1. Initial program 99.7%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. associate-*l* N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 73.9

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

   9. Applied rewrites 73.9%

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

   if 1.7000000000000001e30 < t < 1.0000000000000001e227

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. exp-sqrt N/A

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

      9. sqrt-unprod N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 65.7

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

   7. Applied rewrites 65.7%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 57.4

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

   10. Applied rewrites 57.4%

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

   if 1.0000000000000001e227 < t

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. exp-sqrt N/A

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

      9. sqrt-unprod N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 81.0

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

   7. Applied rewrites 81.0%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 81.0

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

   10. Applied rewrites 81.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{elif}\;t \leq 10^{+227}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \sqrt{2 \cdot \left(z \cdot \mathsf{fma}\left(t, t, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot \mathsf{fma}\left(t, t, 1\right)\right)} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ t_2 := t\_1 \cdot \left(x \cdot 0.5\right)\\ \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{+39}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \cdot 0.5 \leq 2 \cdot 10^{+53}:\\ \;\;\;\;t\_1 \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))) (t_2 (* t_1 (* x 0.5))))
   (if (<= (* x 0.5) -1e+39) t_2 (if (<= (* x 0.5) 2e+53) (* t_1 (- y)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double t_2 = t_1 * (x * 0.5);
	double tmp;
	if ((x * 0.5) <= -1e+39) {
		tmp = t_2;
	} else if ((x * 0.5) <= 2e+53) {
		tmp = t_1 * -y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    t_2 = t_1 * (x * 0.5d0)
    if ((x * 0.5d0) <= (-1d+39)) then
        tmp = t_2
    else if ((x * 0.5d0) <= 2d+53) then
        tmp = t_1 * -y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double t_2 = t_1 * (x * 0.5);
	double tmp;
	if ((x * 0.5) <= -1e+39) {
		tmp = t_2;
	} else if ((x * 0.5) <= 2e+53) {
		tmp = t_1 * -y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	t_2 = t_1 * (x * 0.5)
	tmp = 0
	if (x * 0.5) <= -1e+39:
		tmp = t_2
	elif (x * 0.5) <= 2e+53:
		tmp = t_1 * -y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	t_2 = Float64(t_1 * Float64(x * 0.5))
	tmp = 0.0
	if (Float64(x * 0.5) <= -1e+39)
		tmp = t_2;
	elseif (Float64(x * 0.5) <= 2e+53)
		tmp = Float64(t_1 * Float64(-y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	t_2 = t_1 * (x * 0.5);
	tmp = 0.0;
	if ((x * 0.5) <= -1e+39)
		tmp = t_2;
	elseif ((x * 0.5) <= 2e+53)
		tmp = t_1 * -y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * 0.5), $MachinePrecision], -1e+39], t$95$2, If[LessEqual[N[(x * 0.5), $MachinePrecision], 2e+53], N[(t$95$1 * (-y)), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -9.9999999999999994e38 or 2e53 < (*.f64 x #s(literal 1/2 binary64))

   1. Initial program 99.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. associate-*l* N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 60.1

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

   9. Applied rewrites 60.1%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 50.9%

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

   if -9.9999999999999994e38 < (*.f64 x #s(literal 1/2 binary64)) < 2e53

   1. Initial program 99.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. associate-*l* N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 59.2

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

   9. Applied rewrites 59.2%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 47.3%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 2 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. div-inv N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft-neg-in N/A

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

   7. associate-*l* N/A

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

   8. exp-prod N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 89.5

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

9. Applied rewrites 89.5%

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

10. Final simplification 89.5%

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

Alternative 14: 65.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot \mathsf{fma}\left(t, t, 1\right)\right)} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 3.5e+75)
   (* (sqrt (* z 2.0)) (- (* x 0.5) y))
   (* (sqrt (* 2.0 (* z (fma t t 1.0)))) (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3.5e+75) {
		tmp = sqrt((z * 2.0)) * ((x * 0.5) - y);
	} else {
		tmp = sqrt((2.0 * (z * fma(t, t, 1.0)))) * -y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 3.5e+75)
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(Float64(x * 0.5) - y));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(z * fma(t, t, 1.0)))) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 3.5e+75], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(z * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.4999999999999998e75

   1. Initial program 99.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. associate-*l* N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 70.7

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

   9. Applied rewrites 70.7%

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

   if 3.4999999999999998e75 < t

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. exp-sqrt N/A

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

      9. sqrt-unprod N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 76.1

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

   7. Applied rewrites 76.1%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 61.4

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

   10. Applied rewrites 61.4%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot \mathsf{fma}\left(t, t, 1\right)\right)} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 84.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. exp-sqrt N/A

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

   9. sqrt-unprod N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-exp.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 85.8

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

7. Applied rewrites 85.8%

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

Alternative 16: 80.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. exp-sqrt N/A

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

   9. sqrt-unprod N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-exp.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 82.5

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

7. Applied rewrites 82.5%

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

8. Final simplification 82.5%

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

Alternative 17: 57.1% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. div-inv N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft-neg-in N/A

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

   7. associate-*l* N/A

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

   8. exp-prod N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in t around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 59.6

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

9. Applied rewrites 59.6%

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

10. Final simplification 59.6%

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

Alternative 18: 30.5% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-exp.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. div-inv N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft-neg-in N/A

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

   7. associate-*l* N/A

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

   8. exp-prod N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in t around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 59.6

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

9. Applied rewrites 59.6%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 32.2%

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

13. Final simplification 32.2%

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

Developer Target 1: 99.4% accurate, 0.6× 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 2024219 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* (* (- (* x 1/2) y) (sqrt (* z 2))) (pow (exp 1) (/ (* t t) 2))))

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