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

Percentage Accurate: 99.4% → 99.4%

Time: 21.7s

Alternatives: 18

Speedup: 1.1×

• 44.3% 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

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{0 - t}\right)}^{\left(t \cdot -0.5\right)} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (exp (- 0.0 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((0.0 - 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((0.0d0 - 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((0.0 - 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((0.0 - 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(0.0 - t)) ^ Float64(t * -0.5)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * (exp((0.0 - 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[N[(0.0 - t), $MachinePrecision]], $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. frac-2neg N/A

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

   2. div-inv N/A

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

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

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

   4. associate-*l* N/A

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

   5. exp-prod N/A

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

   6. pow-lowering-pow.f64 N/A

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

   7. exp-lowering-exp.f64 N/A

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

   8. neg-sub0 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{\color{blue}{0 - t}}\right)}^{\left(t \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)} \]

   9. --lowering--.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{\color{blue}{0 - t}}\right)}^{\left(t \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{0 - t}\right)}^{\color{blue}{\left(t \cdot \frac{1}{\mathsf{neg}\left(2\right)}\right)}} \]

   11. metadata-eval N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{0 - t}\right)}^{\left(t \cdot \frac{1}{\color{blue}{-2}}\right)} \]

   12. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 2: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * sqrt(((z * 2.0) * pow(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(((z * 2.0d0) * (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(((z * 2.0) * Math.pow(Math.exp(t), t)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. *-lowering-*.f64 N/A

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

   3. --lowering--.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. exp-sqrt N/A

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

   6. sqrt-unprod N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. exp-lowering-exp.f64 N/A

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

   11. *-lowering-*.f64 99.8

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

4. Applied egg-rr 99.8%

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

   1. exp-prod N/A

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

   2. pow-lowering-pow.f64 N/A

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

   3. exp-lowering-exp.f64 99.8

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

6. Applied egg-rr 99.8%

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

Alternative 3: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * sqrt(((z * 2.0) * 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(((z * 2.0d0) * 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(((z * 2.0) * Math.exp((t * t))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. *-lowering-*.f64 N/A

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

   3. --lowering--.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. exp-sqrt N/A

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

   6. sqrt-unprod N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. exp-lowering-exp.f64 N/A

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

   11. *-lowering-*.f64 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 4: 96.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 100:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t \cdot t, 0.3333333333333333, 1\right), 2\right), t \cdot t, z \cdot 2\right)}\\ \mathbf{elif}\;t \cdot t \leq 10^{+47}:\\ \;\;\;\;\frac{t\_2 \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(0 - y, \frac{y}{x \cdot x}, 0.25\right)\right)\right)}{\mathsf{fma}\left(x, 0.5, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot t\_2\right) \cdot \mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* z 2.0))))
   (if (<= (* t t) 100.0)
     (*
      t_1
      (sqrt
       (fma
        (* z (fma t (* t (fma (* t t) 0.3333333333333333 1.0)) 2.0))
        (* t t)
        (* z 2.0))))
     (if (<= (* t t) 1e+47)
       (/ (* t_2 (* x (* x (fma (- 0.0 y) (/ y (* x x)) 0.25)))) (fma x 0.5 y))
       (*
        (* t_1 t_2)
        (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 t_2 = sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 100.0) {
		tmp = t_1 * sqrt(fma((z * fma(t, (t * fma((t * t), 0.3333333333333333, 1.0)), 2.0)), (t * t), (z * 2.0)));
	} else if ((t * t) <= 1e+47) {
		tmp = (t_2 * (x * (x * fma((0.0 - y), (y / (x * x)), 0.25)))) / fma(x, 0.5, y);
	} else {
		tmp = (t_1 * t_2) * 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)
	t_2 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (Float64(t * t) <= 100.0)
		tmp = Float64(t_1 * sqrt(fma(Float64(z * fma(t, Float64(t * fma(Float64(t * t), 0.3333333333333333, 1.0)), 2.0)), Float64(t * t), Float64(z * 2.0))));
	elseif (Float64(t * t) <= 1e+47)
		tmp = Float64(Float64(t_2 * Float64(x * Float64(x * fma(Float64(0.0 - y), Float64(y / Float64(x * x)), 0.25)))) / fma(x, 0.5, y));
	else
		tmp = Float64(Float64(t_1 * t_2) * fma(t, Float64(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]}, Block[{t$95$2 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 100.0], N[(t$95$1 * N[Sqrt[N[(N[(z * N[(t * N[(t * N[(N[(t * t), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(z * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 1e+47], N[(N[(t$95$2 * N[(x * N[(x * N[(N[(0.0 - y), $MachinePrecision] * N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.5 + y), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * t$95$2), $MachinePrecision] * N[(t * N[(t * N[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.020833333333333332 + 0.125), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 100

   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. associate-*l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. exp-sqrt N/A

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

      6. sqrt-unprod N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. *-lowering-*.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

         \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{{t}^{2} \cdot \left(2 \cdot z + {t}^{2} \cdot \left(z + \frac{1}{3} \cdot \left({t}^{2} \cdot z\right)\right)\right) + 2 \cdot z}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left({t}^{2}, 2 \cdot z + {t}^{2} \cdot \left(z + \frac{1}{3} \cdot \left({t}^{2} \cdot z\right)\right), 2 \cdot z\right)}} \]

      3. unpow2 N/A

         \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{t \cdot t}, 2 \cdot z + {t}^{2} \cdot \left(z + \frac{1}{3} \cdot \left({t}^{2} \cdot z\right)\right), 2 \cdot z\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{t \cdot t}, 2 \cdot z + {t}^{2} \cdot \left(z + \frac{1}{3} \cdot \left({t}^{2} \cdot z\right)\right), 2 \cdot z\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(t \cdot t, \color{blue}{{t}^{2} \cdot \left(z + \frac{1}{3} \cdot \left({t}^{2} \cdot z\right)\right) + 2 \cdot z}, 2 \cdot z\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(t \cdot t, \color{blue}{\mathsf{fma}\left({t}^{2}, z + \frac{1}{3} \cdot \left({t}^{2} \cdot z\right), 2 \cdot z\right)}, 2 \cdot z\right)} \]

      7. unpow2 N/A

         \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, z + \frac{1}{3} \cdot \left({t}^{2} \cdot z\right), 2 \cdot z\right), 2 \cdot z\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, z + \frac{1}{3} \cdot \left({t}^{2} \cdot z\right), 2 \cdot z\right), 2 \cdot z\right)} \]

      9. associate-*r* N/A

         \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, z + \color{blue}{\left(\frac{1}{3} \cdot {t}^{2}\right) \cdot z}, 2 \cdot z\right), 2 \cdot z\right)} \]

      10. distribute-rgt1-in N/A

          \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \color{blue}{\left(\frac{1}{3} \cdot {t}^{2} + 1\right) \cdot z}, 2 \cdot z\right), 2 \cdot z\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \color{blue}{\left(\frac{1}{3} \cdot {t}^{2} + 1\right) \cdot z}, 2 \cdot z\right), 2 \cdot z\right)} \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \left(\color{blue}{{t}^{2} \cdot \frac{1}{3}} + 1\right) \cdot z, 2 \cdot z\right), 2 \cdot z\right)} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \color{blue}{\mathsf{fma}\left({t}^{2}, \frac{1}{3}, 1\right)} \cdot z, 2 \cdot z\right), 2 \cdot z\right)} \]

      14. unpow2 N/A

          \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{3}, 1\right) \cdot z, 2 \cdot z\right), 2 \cdot z\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{3}, 1\right) \cdot z, 2 \cdot z\right), 2 \cdot z\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \frac{1}{3}, 1\right) \cdot z, \color{blue}{2 \cdot z}\right), 2 \cdot z\right)} \]

      17. *-lowering-*.f64 98.8

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

   7. Simplified 98.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.3333333333333333, 1\right) \cdot z, 2 \cdot z\right), 2 \cdot z\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right) \cdot z\right) + 2 \cdot z\right) \cdot \left(t \cdot t\right)} + 2 \cdot z} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\left(t \cdot t\right) \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right) \cdot z\right) + 2 \cdot z, t \cdot t, 2 \cdot z\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\left(\left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right) \cdot z} + 2 \cdot z, t \cdot t, 2 \cdot z\right)} \]

      4. distribute-rgt-out N/A

         \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{z \cdot \left(\left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right) + 2\right)}, t \cdot t, 2 \cdot z\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{z \cdot \left(\left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right) + 2\right)}, t \cdot t, 2 \cdot z\right)} \]

      6. associate-*l* N/A

         \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(z \cdot \left(\color{blue}{t \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)} + 2\right), t \cdot t, 2 \cdot z\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(z \cdot \color{blue}{\mathsf{fma}\left(t, t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right), 2\right)}, t \cdot t, 2 \cdot z\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(t, \color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)}, 2\right), t \cdot t, 2 \cdot z\right)} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(t, t \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, \frac{1}{3}, 1\right)}, 2\right), t \cdot t, 2 \cdot z\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{3}, 1\right), 2\right), t \cdot t, 2 \cdot z\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{3}, 1\right), 2\right), \color{blue}{t \cdot t}, 2 \cdot z\right)} \]

      12. *-commutative N/A

          \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{3}, 1\right), 2\right), t \cdot t, \color{blue}{z \cdot 2}\right)} \]

      13. *-lowering-*.f64 98.8

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

   9. Applied egg-rr 98.8%

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

   if 100 < (*.f64 t t) < 1e47

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

   5. Simplified 31.8%

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

      1. *-rgt-identity N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 72.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 0.25 \cdot x, 0 - y \cdot y\right) \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, 0.5, y\right)}} \]

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{4} + -1 \cdot \frac{{y}^{2}}{{x}^{2}}\right)\right) \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, \frac{1}{2}, y\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{4} + -1 \cdot \frac{{y}^{2}}{{x}^{2}}\right)\right)\right)} \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, \frac{1}{2}, y\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{4} + -1 \cdot \frac{{y}^{2}}{{x}^{2}}\right)\right)\right)} \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, \frac{1}{2}, y\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{4} + -1 \cdot \frac{{y}^{2}}{{x}^{2}}\right)\right)}\right) \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, \frac{1}{2}, y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \color{blue}{\left(-1 \cdot \frac{{y}^{2}}{{x}^{2}} + \frac{1}{4}\right)}\right)\right) \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, \frac{1}{2}, y\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{{y}^{2}}{{x}^{2}}\right)\right)} + \frac{1}{4}\right)\right)\right) \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, \frac{1}{2}, y\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\left(\mathsf{neg}\left(\frac{\color{blue}{y \cdot y}}{{x}^{2}}\right)\right) + \frac{1}{4}\right)\right)\right) \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, \frac{1}{2}, y\right)} \]

      8. associate-/l* N/A

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\left(\mathsf{neg}\left(\color{blue}{y \cdot \frac{y}{{x}^{2}}}\right)\right) + \frac{1}{4}\right)\right)\right) \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, \frac{1}{2}, y\right)} \]

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

         \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{y}{{x}^{2}}} + \frac{1}{4}\right)\right)\right) \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, \frac{1}{2}, y\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\color{blue}{\left(-1 \cdot y\right)} \cdot \frac{y}{{x}^{2}} + \frac{1}{4}\right)\right)\right) \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, \frac{1}{2}, y\right)} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(-1 \cdot y, \frac{y}{{x}^{2}}, \frac{1}{4}\right)}\right)\right) \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, \frac{1}{2}, y\right)} \]

      12. mul-1-neg N/A

          \[\leadsto \frac{\left(x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, \frac{y}{{x}^{2}}, \frac{1}{4}\right)\right)\right) \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, \frac{1}{2}, y\right)} \]

      13. neg-sub0 N/A

          \[\leadsto \frac{\left(x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{0 - y}, \frac{y}{{x}^{2}}, \frac{1}{4}\right)\right)\right) \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, \frac{1}{2}, y\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \frac{\left(x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{0 - y}, \frac{y}{{x}^{2}}, \frac{1}{4}\right)\right)\right) \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, \frac{1}{2}, y\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \frac{\left(x \cdot \left(x \cdot \mathsf{fma}\left(0 - y, \color{blue}{\frac{y}{{x}^{2}}}, \frac{1}{4}\right)\right)\right) \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, \frac{1}{2}, y\right)} \]

      16. unpow2 N/A

          \[\leadsto \frac{\left(x \cdot \left(x \cdot \mathsf{fma}\left(0 - y, \frac{y}{\color{blue}{x \cdot x}}, \frac{1}{4}\right)\right)\right) \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, \frac{1}{2}, y\right)} \]

      17. *-lowering-*.f64 93.2

          \[\leadsto \frac{\left(x \cdot \left(x \cdot \mathsf{fma}\left(0 - y, \frac{y}{\color{blue}{x \cdot x}}, 0.25\right)\right)\right) \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, 0.5, y\right)} \]

   10. Simplified 93.2%

       \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot \mathsf{fma}\left(0 - y, \frac{y}{x \cdot x}, 0.25\right)\right)\right)} \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, 0.5, y\right)} \]

   if 1e47 < (*.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. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 97.7

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

   5. Simplified 97.7%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 100:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t \cdot t, 0.3333333333333333, 1\right), 2\right), t \cdot t, z \cdot 2\right)}\\ \mathbf{elif}\;t \cdot t \leq 10^{+47}:\\ \;\;\;\;\frac{\sqrt{z \cdot 2} \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(0 - y, \frac{y}{x \cdot x}, 0.25\right)\right)\right)}{\mathsf{fma}\left(x, 0.5, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{z \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (sqrt
   (*
    z
    (fma (* t t) (fma (* t t) (fma (* t t) 0.16666666666666666 0.5) 1.0) 1.0)))
  (* (- (* x 0.5) y) (sqrt 2.0))))

C

double code(double x, double y, double z, double t) {
	return sqrt((z * fma((t * t), fma((t * t), fma((t * t), 0.16666666666666666, 0.5), 1.0), 1.0))) * (((x * 0.5) - y) * sqrt(2.0));
}

Julia

function code(x, y, z, t)
	return Float64(sqrt(Float64(z * fma(Float64(t * t), fma(Float64(t * t), fma(Float64(t * t), 0.16666666666666666, 0.5), 1.0), 1.0))) * Float64(Float64(Float64(x * 0.5) - y) * sqrt(2.0)))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(z * N[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. pow1/2 N/A

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

   4. unpow-prod-down N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. *-lowering-*.f64 N/A

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

   8. exp-sqrt N/A

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

   9. pow1/2 N/A

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

   10. sqrt-unprod N/A

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

   11. sqrt-lowering-sqrt.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. exp-lowering-exp.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. *-commutative N/A

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

   16. *-lowering-*.f64 N/A

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

   17. --lowering--.f64 N/A

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

   18. *-lowering-*.f64 N/A

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

   19. pow1/2 N/A

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

   20. sqrt-lowering-sqrt.f64 99.7

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left({t}^{2}, 1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {t}^{2}\right), 1\right)} \cdot z} \cdot \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{2}\right) \]

   3. unpow2 N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{t \cdot t}, 1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {t}^{2}\right), 1\right) \cdot z} \cdot \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{2}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{t \cdot t}, 1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {t}^{2}\right), 1\right) \cdot z} \cdot \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{2}\right) \]

   5. +-commutative N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(t \cdot t, \color{blue}{{t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {t}^{2}\right) + 1}, 1\right) \cdot z} \cdot \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{2}\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(t \cdot t, \color{blue}{\mathsf{fma}\left({t}^{2}, \frac{1}{2} + \frac{1}{6} \cdot {t}^{2}, 1\right)}, 1\right) \cdot z} \cdot \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{2}\right) \]

   7. unpow2 N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{2} + \frac{1}{6} \cdot {t}^{2}, 1\right), 1\right) \cdot z} \cdot \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{2}\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{1}{2} + \frac{1}{6} \cdot {t}^{2}, 1\right), 1\right) \cdot z} \cdot \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{2}\right) \]

   9. +-commutative N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \color{blue}{\frac{1}{6} \cdot {t}^{2} + \frac{1}{2}}, 1\right), 1\right) \cdot z} \cdot \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{2}\right) \]

   10. *-commutative N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \color{blue}{{t}^{2} \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot z} \cdot \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{2}\right) \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \color{blue}{\mathsf{fma}\left({t}^{2}, \frac{1}{6}, \frac{1}{2}\right)}, 1\right), 1\right) \cdot z} \cdot \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{2}\right) \]

   12. unpow2 N/A

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

   13. *-lowering-*.f64 94.9

       \[\leadsto \sqrt{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{t \cdot t}, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot z} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right) \]

7. Simplified 94.9%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot z} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right) \]

8. Final simplification 94.9%

   \[\leadsto \sqrt{z \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right) \]
9. Add Preprocessing

Alternative 6: 94.5% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (- (* x 0.5) y) (sqrt (* z 2.0)))
  (fma
   t
   (* t (fma (* t t) (fma (* t t) 0.020833333333333332 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), fma((t * t), 0.020833333333333332, 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(Float64(t * t), fma(Float64(t * t), 0.020833333333333332, 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[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.020833333333333332 + 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

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. +-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 94.6

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

5. Simplified 94.6%

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

Alternative 7: 93.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. 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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 92.3

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

5. Simplified 92.3%

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

6. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-in N/A

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

8. Simplified 93.2%

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

9. Final simplification 93.2%

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

Alternative 8: 92.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \cdot t \leq 5000000000000:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(t \cdot t, z \cdot \mathsf{fma}\left(t, t, 2\right), z \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{z \cdot 2}\right) \cdot \left(0.125 \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot t\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* t t) 5000000000000.0)
     (* t_1 (sqrt (fma (* t t) (* z (fma t t 2.0)) (* z 2.0))))
     (* (* t_1 (sqrt (* z 2.0))) (* 0.125 (* (* t t) (* t t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((t * t) <= 5000000000000.0) {
		tmp = t_1 * sqrt(fma((t * t), (z * fma(t, t, 2.0)), (z * 2.0)));
	} else {
		tmp = (t_1 * sqrt((z * 2.0))) * (0.125 * ((t * t) * (t * t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(t * t) <= 5000000000000.0)
		tmp = Float64(t_1 * sqrt(fma(Float64(t * t), Float64(z * fma(t, t, 2.0)), Float64(z * 2.0))));
	else
		tmp = Float64(Float64(t_1 * sqrt(Float64(z * 2.0))) * Float64(0.125 * Float64(Float64(t * t) * Float64(t * t))));
	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], 5000000000000.0], N[(t$95$1 * N[Sqrt[N[(N[(t * t), $MachinePrecision] * N[(z * N[(t * t + 2.0), $MachinePrecision]), $MachinePrecision] + N[(z * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(N[(t * t), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   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. associate-*l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. exp-sqrt N/A

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

      6. sqrt-unprod N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. *-lowering-*.f64 99.5

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

   4. Applied egg-rr 99.5%

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

      1. exp-prod N/A

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

      2. pow-lowering-pow.f64 N/A

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

      3. exp-lowering-exp.f64 99.5

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. distribute-rgt-out N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 96.4

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

   9. Simplified 96.4%

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

   if 5e12 < (*.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. 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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 89.2

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

   5. Simplified 89.2%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 89.2

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

   8. Simplified 89.2%

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

   9. Taylor expanded in t around inf

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

       1. *-lowering-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 89.2

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

   11. Simplified 89.2%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 5000000000000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\mathsf{fma}\left(t \cdot t, z \cdot \mathsf{fma}\left(t, t, 2\right), z \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(0.125 \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot t\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;z \cdot 2 \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(0.5, t \cdot t, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\left(z \cdot 2\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* z 2.0) 5e+16)
     (* (* t_1 (sqrt (* z 2.0))) (fma 0.5 (* t t) 1.0))
     (* t_1 (sqrt (* (* z 2.0) (fma t t 1.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((z * 2.0) <= 5e+16) {
		tmp = (t_1 * sqrt((z * 2.0))) * fma(0.5, (t * t), 1.0);
	} else {
		tmp = t_1 * sqrt(((z * 2.0) * fma(t, t, 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(z * 2.0) <= 5e+16)
		tmp = Float64(Float64(t_1 * sqrt(Float64(z * 2.0))) * fma(0.5, Float64(t * t), 1.0));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(z * 2.0) * fma(t, t, 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[(z * 2.0), $MachinePrecision], 5e+16], N[(N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 2 binary64)) < 5e16

   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. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 82.0

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

   5. Simplified 82.0%

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

   if 5e16 < (*.f64 z #s(literal 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. associate-*l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. exp-sqrt N/A

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

      6. sqrt-unprod N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. *-lowering-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. accelerator-lowering-fma.f64 95.8

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

   7. Simplified 95.8%

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

Alternative 10: 93.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{z \cdot 2} \cdot \left(\mathsf{fma}\left(t, 0.125 \cdot \left(t \cdot \left(t \cdot t\right)\right), 1\right) \cdot \mathsf{fma}\left(x, 0.5, 0 - y\right)\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (sqrt (* z 2.0))
  (* (fma t (* 0.125 (* t (* t t))) 1.0) (fma x 0.5 (- 0.0 y)))))

C

double code(double x, double y, double z, double t) {
	return sqrt((z * 2.0)) * (fma(t, (0.125 * (t * (t * t))), 1.0) * fma(x, 0.5, (0.0 - y)));
}

Julia

function code(x, y, z, t)
	return Float64(sqrt(Float64(z * 2.0)) * Float64(fma(t, Float64(0.125 * Float64(t * Float64(t * t))), 1.0) * fma(x, 0.5, Float64(0.0 - y))))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(t * N[(0.125 * N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * 0.5 + N[(0.0 - y), $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. 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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 92.3

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

5. Simplified 92.3%

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

6. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 92.1

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

8. Simplified 92.1%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. associate-*r* N/A

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

   7. cube-unmult N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 N/A

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

   10. cube-unmult N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. sub-neg N/A

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

   14. accelerator-lowering-fma.f64 N/A

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

   15. neg-sub0 N/A

       \[\leadsto \left(\mathsf{fma}\left(t, \frac{1}{8} \cdot \left(t \cdot \left(t \cdot t\right)\right), 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{0 - y}\right)\right) \cdot \sqrt{z \cdot 2} \]

   16. --lowering--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(t, \frac{1}{8} \cdot \left(t \cdot \left(t \cdot t\right)\right), 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{0 - y}\right)\right) \cdot \sqrt{z \cdot 2} \]

   17. *-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(t, \frac{1}{8} \cdot \left(t \cdot \left(t \cdot t\right)\right), 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, 0 - y\right)\right) \cdot \sqrt{\color{blue}{2 \cdot z}} \]

   18. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(t, \frac{1}{8} \cdot \left(t \cdot \left(t \cdot t\right)\right), 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, 0 - y\right)\right) \cdot \color{blue}{\sqrt{2 \cdot z}} \]

   19. *-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(t, \frac{1}{8} \cdot \left(t \cdot \left(t \cdot t\right)\right), 1\right) \cdot \mathsf{fma}\left(x, \frac{1}{2}, 0 - y\right)\right) \cdot \sqrt{\color{blue}{z \cdot 2}} \]

   20. *-lowering-*.f64 93.2

       \[\leadsto \left(\mathsf{fma}\left(t, 0.125 \cdot \left(t \cdot \left(t \cdot t\right)\right), 1\right) \cdot \mathsf{fma}\left(x, 0.5, 0 - y\right)\right) \cdot \sqrt{\color{blue}{z \cdot 2}} \]

10. Applied egg-rr 93.2%

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(t, 0.125 \cdot \left(t \cdot \left(t \cdot t\right)\right), 1\right) \cdot \mathsf{fma}\left(x, 0.5, 0 - y\right)\right) \cdot \sqrt{z \cdot 2}} \]

11. Final simplification 93.2%

    \[\leadsto \sqrt{z \cdot 2} \cdot \left(\mathsf{fma}\left(t, 0.125 \cdot \left(t \cdot \left(t \cdot t\right)\right), 1\right) \cdot \mathsf{fma}\left(x, 0.5, 0 - y\right)\right) \]
12. Add Preprocessing

Alternative 11: 92.1% accurate, 2.7× speedup

Math

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

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(Float64(t * 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[(N[(t * t), $MachinePrecision] * 0.125 + 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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 92.3

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

5. Simplified 92.3%

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

Alternative 12: 91.8% 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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 92.3

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

5. Simplified 92.3%

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

6. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 92.1

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

8. Simplified 92.1%

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

Alternative 13: 89.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. --lowering--.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. exp-sqrt N/A

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

   6. sqrt-unprod N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. exp-lowering-exp.f64 N/A

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

   11. *-lowering-*.f64 99.8

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

4. Applied egg-rr 99.8%

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

   1. exp-prod N/A

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

   2. pow-lowering-pow.f64 N/A

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

   3. exp-lowering-exp.f64 99.8

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. distribute-rgt-out N/A

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

   6. *-lowering-*.f64 N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. *-lowering-*.f64 90.0

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

9. Simplified 90.0%

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

10. Final simplification 90.0%

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

Alternative 14: 42.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ t_2 := t\_1 \cdot \mathsf{fma}\left(x, 0.5, y\right)\\ \mathbf{if}\;x \cdot 0.5 \leq -4 \cdot 10^{-43}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \cdot 0.5 \leq 5 \cdot 10^{+72}:\\ \;\;\;\;t\_1 \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))) (t_2 (* t_1 (fma x 0.5 y))))
   (if (<= (* x 0.5) -4e-43)
     t_2
     (if (<= (* x 0.5) 5e+72) (* t_1 (- 0.0 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 * fma(x, 0.5, y);
	double tmp;
	if ((x * 0.5) <= -4e-43) {
		tmp = t_2;
	} else if ((x * 0.5) <= 5e+72) {
		tmp = t_1 * (0.0 - 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 * fma(x, 0.5, y))
	tmp = 0.0
	if (Float64(x * 0.5) <= -4e-43)
		tmp = t_2;
	elseif (Float64(x * 0.5) <= 5e+72)
		tmp = Float64(t_1 * Float64(0.0 - y));
	else
		tmp = t_2;
	end
	return 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 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * 0.5), $MachinePrecision], -4e-43], t$95$2, If[LessEqual[N[(x * 0.5), $MachinePrecision], 5e+72], N[(t$95$1 * N[(0.0 - y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -4.00000000000000031e-43 or 4.99999999999999992e72 < (*.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. Taylor expanded in t around 0

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

   5. Simplified 58.5%

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

      1. *-rgt-identity N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 0.25 \cdot x, 0 - y \cdot y\right) \cdot \sqrt{z \cdot 2}}{\mathsf{fma}\left(x, 0.5, y\right)}} \]

   9. Applied egg-rr 48.0%

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

   if -4.00000000000000031e-43 < (*.f64 x #s(literal 1/2 binary64)) < 4.99999999999999992e72

   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. Taylor expanded in t around 0

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

   5. Simplified 51.3%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \left(\color{blue}{\left(0 - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]

      3. --lowering--.f64 41.2

         \[\leadsto \left(\color{blue}{\left(0 - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]

   8. Simplified 41.2%

      \[\leadsto \left(\color{blue}{\left(0 - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]

   10. Applied egg-rr 41.2%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -4 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \mathsf{fma}\left(x, 0.5, y\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 5 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \mathsf{fma}\left(x, 0.5, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 84.3% accurate, 3.8× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. --lowering--.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. exp-sqrt N/A

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

   6. sqrt-unprod N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. exp-lowering-exp.f64 N/A

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

   11. *-lowering-*.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-fma.f64 85.5

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

7. Simplified 85.5%

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

Alternative 16: 57.1% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

5. Simplified 54.6%

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

   1. *-rgt-identity N/A

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

   2. *-lowering-*.f64 N/A

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

   3. --lowering--.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. *-lowering-*.f64 54.6

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

7. Applied egg-rr 54.6%

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

Alternative 17: 29.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{z \cdot 2} \cdot \left(0 - y\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (sqrt (* z 2.0)) (- 0.0 y)))

C

double code(double x, double y, double z, double t) {
	return sqrt((z * 2.0)) * (0.0 - 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)) * (0.0d0 - y)
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.sqrt((z * 2.0)) * (0.0 - y);
}

Python

def code(x, y, z, t):
	return math.sqrt((z * 2.0)) * (0.0 - y)

Julia

function code(x, y, z, t)
	return Float64(sqrt(Float64(z * 2.0)) * Float64(0.0 - y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = sqrt((z * 2.0)) * (0.0 - y);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(0.0 - 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. Taylor expanded in t around 0

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

5. Simplified 54.6%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \left(\color{blue}{\left(0 - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]

   3. --lowering--.f64 28.3

      \[\leadsto \left(\color{blue}{\left(0 - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]

8. Simplified 28.3%

   \[\leadsto \left(\color{blue}{\left(0 - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]

10. Applied egg-rr 28.3%

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

11. Final simplification 28.3%

    \[\leadsto \sqrt{z \cdot 2} \cdot \left(0 - y\right) \]
12. Add Preprocessing

Alternative 18: 2.4% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

5. Simplified 54.6%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \left(\color{blue}{\left(0 - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]

   3. --lowering--.f64 28.3

      \[\leadsto \left(\color{blue}{\left(0 - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]

8. Simplified 28.3%

   \[\leadsto \left(\color{blue}{\left(0 - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]
9. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto \color{blue}{\left(0 - y\right) \cdot \sqrt{z \cdot 2}} \]

   2. sub0-neg N/A

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

   3. *-commutative N/A

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

   4. sub0-neg N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \color{blue}{\left(0 - y\right)} \]

   5. flip-- N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \color{blue}{\frac{0 \cdot 0 - y \cdot y}{0 + y}} \]

10. Applied egg-rr 2.7%

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

11. Final simplification 2.7%

    \[\leadsto y \cdot \sqrt{z \cdot 2} \]
12. 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 2024195 
(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))))