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

Percentage Accurate: 99.5% → 99.8%

Time: 15.4s

Alternatives: 15

Speedup: 0.6×

• 44.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-*.f64 N/A

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

   11. unpow-prod-down N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. pow1/2 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. exp-sqrt N/A

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

   18. sqrt-unprod N/A

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

   19. pow1/2 N/A

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

   20. sqrt-unprod N/A

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

   21. lower-sqrt.f64 N/A

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

4. Applied rewrites 99.8%

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

Alternative 2: 74.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-*.f64 N/A

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

   11. unpow-prod-down N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. pow1/2 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. exp-sqrt N/A

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

   18. sqrt-unprod N/A

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

   19. pow1/2 N/A

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

   20. sqrt-unprod N/A

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

   21. lower-sqrt.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. lower-+.f64 76.2

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

7. Applied rewrites 76.2%

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

Alternative 3: 95.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. 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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 95.4

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

5. Applied rewrites 95.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 96.1

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

7. Applied rewrites 96.1%

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

9. Applied rewrites 96.1%

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

Alternative 4: 95.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. 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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 95.4

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

5. Applied rewrites 95.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

7. Applied rewrites 96.1%

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

Alternative 5: 95.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. 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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 95.4

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

5. Applied rewrites 95.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 96.1

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

7. Applied rewrites 96.1%

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

8. Taylor expanded in t around inf

   \[\leadsto \left(x \cdot \frac{1}{2} - y\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48} \cdot {t}^{2}, t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right) \cdot \sqrt{z \cdot 2}\right) \]
9. Step-by-step derivation

10. Applied rewrites 96.1%

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

Alternative 6: 94.6% accurate, 2.3× speedup

Math

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

FPCore

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

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * fma(fma(Float64(0.020833333333333332 * Float64(t * t)), Float64(t * t), 0.5), Float64(t * t), 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[(N[(N[(0.020833333333333332 * N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. 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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 95.4

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

5. Applied rewrites 95.4%

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

6. Taylor expanded in t around inf

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48} \cdot {t}^{2}, t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right) \]
7. Step-by-step derivation

8. Applied rewrites 95.3%

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

9. Final simplification 95.3%

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

Alternative 7: 87.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 2 \leq 10^{+143}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot \sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{\mathsf{fma}\left(t \cdot t, z, z\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 2.0) 1e+143)
   (* (- (* x 0.5) y) (* (fma (* t t) 0.5 1.0) (sqrt (* z 2.0))))
   (* (- (* 0.5 x) y) (sqrt (* (fma (* t t) z z) 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 2.0) <= 1e+143) {
		tmp = ((x * 0.5) - y) * (fma((t * t), 0.5, 1.0) * sqrt((z * 2.0)));
	} else {
		tmp = ((0.5 * x) - y) * sqrt((fma((t * t), z, z) * 2.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 2.0) <= 1e+143)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * Float64(fma(Float64(t * t), 0.5, 1.0) * sqrt(Float64(z * 2.0))));
	else
		tmp = Float64(Float64(Float64(0.5 * x) - y) * sqrt(Float64(fma(Float64(t * t), z, z) * 2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 2.0), $MachinePrecision], 1e+143], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[(N[(t * t), $MachinePrecision] * z + z), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 2 binary64)) < 1e143

   1. Initial program 99.3%

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

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.8

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 84.3

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

   10. Applied rewrites 84.3%

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

   if 1e143 < (*.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. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. pow1/2 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. exp-sqrt N/A

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

      18. sqrt-unprod N/A

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

      19. pow1/2 N/A

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

      20. sqrt-unprod N/A

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

      21. lower-sqrt.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 98.1

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

   7. Applied rewrites 98.1%

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

Alternative 8: 86.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 2 \leq 10^{+143}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{\mathsf{fma}\left(t \cdot t, z, z\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 2.0) 1e+143)
   (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (fma (* t t) 0.5 1.0))
   (* (- (* 0.5 x) y) (sqrt (* (fma (* t t) z z) 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 2.0) <= 1e+143) {
		tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * fma((t * t), 0.5, 1.0);
	} else {
		tmp = ((0.5 * x) - y) * sqrt((fma((t * t), z, z) * 2.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 2.0) <= 1e+143)
		tmp = Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * fma(Float64(t * t), 0.5, 1.0));
	else
		tmp = Float64(Float64(Float64(0.5 * x) - y) * sqrt(Float64(fma(Float64(t * t), z, z) * 2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 2.0), $MachinePrecision], 1e+143], N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[(N[(t * t), $MachinePrecision] * z + z), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 2 binary64)) < 1e143

   1. Initial program 99.3%

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if 1e143 < (*.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. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. pow1/2 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. exp-sqrt N/A

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

      18. sqrt-unprod N/A

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

      19. pow1/2 N/A

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

      20. sqrt-unprod N/A

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

      21. lower-sqrt.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 98.1

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

   7. Applied rewrites 98.1%

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

Alternative 9: 93.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. 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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 93.5

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

5. Applied rewrites 93.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

7. Applied rewrites 94.6%

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

Alternative 10: 92.7% accurate, 2.7× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. 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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 93.5

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

5. Applied rewrites 93.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower-*.f64 94.2

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 94.2

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

7. Applied rewrites 94.2%

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

8. Taylor expanded in t around inf

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

10. Applied rewrites 93.9%

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

Alternative 11: 92.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-*.f64 N/A

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

   11. unpow-prod-down N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. pow1/2 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. exp-sqrt N/A

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

   18. sqrt-unprod N/A

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

   19. pow1/2 N/A

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

   20. sqrt-unprod N/A

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

   21. lower-sqrt.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. lower-+.f64 76.2

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

7. Applied rewrites 76.2%

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

8. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. distribute-rgt-out N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. unpow2 N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 93.5

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

10. Applied rewrites 93.5%

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

Alternative 12: 75.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot t \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \sqrt{\mathsf{fma}\left(t \cdot t, z, z\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* t t) 5.8e-7)
   (* (- (* 0.5 x) y) (sqrt (* 2.0 z)))
   (* (- y) (sqrt (* (fma (* t t) z z) 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * t) <= 5.8e-7) {
		tmp = ((0.5 * x) - y) * sqrt((2.0 * z));
	} else {
		tmp = -y * sqrt((fma((t * t), z, z) * 2.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(t * t) <= 5.8e-7)
		tmp = Float64(Float64(Float64(0.5 * x) - y) * sqrt(Float64(2.0 * z)));
	else
		tmp = Float64(Float64(-y) * sqrt(Float64(fma(Float64(t * t), z, z) * 2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(t * t), $MachinePrecision], 5.8e-7], N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-y) * N[Sqrt[N[(N[(N[(t * t), $MachinePrecision] * z + z), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 5.7999999999999995e-7

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. pow1/2 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. exp-sqrt N/A

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

      18. sqrt-unprod N/A

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

      19. pow1/2 N/A

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

      20. sqrt-unprod N/A

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

      21. lower-sqrt.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 99.4

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

   7. Applied rewrites 99.4%

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

   if 5.7999999999999995e-7 < (*.f64 t t)

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. pow1/2 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. exp-sqrt N/A

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

      18. sqrt-unprod N/A

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

      19. pow1/2 N/A

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

      20. sqrt-unprod N/A

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

      21. lower-sqrt.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 10.6

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

   7. Applied rewrites 10.6%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 7.1

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

   10. Applied rewrites 7.1%

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

   11. Taylor expanded in t around 0

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

       1. distribute-lft-out N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 51.7

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

   13. Applied rewrites 51.7%

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

Alternative 13: 84.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-*.f64 N/A

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

   11. unpow-prod-down N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. pow1/2 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. exp-sqrt N/A

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

   18. sqrt-unprod N/A

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

   19. pow1/2 N/A

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

   20. sqrt-unprod N/A

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

   21. lower-sqrt.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 84.8

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

7. Applied rewrites 84.8%

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

Alternative 14: 56.7% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-*.f64 N/A

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

   11. unpow-prod-down N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. pow1/2 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. exp-sqrt N/A

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

   18. sqrt-unprod N/A

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

   19. pow1/2 N/A

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

   20. sqrt-unprod N/A

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

   21. lower-sqrt.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. lower-*.f64 55.7

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

7. Applied rewrites 55.7%

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

Alternative 15: 30.1% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-*.f64 N/A

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

   11. unpow-prod-down N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. pow1/2 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. exp-sqrt N/A

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

   18. sqrt-unprod N/A

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

   19. pow1/2 N/A

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

   20. sqrt-unprod N/A

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

   21. lower-sqrt.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. lower-*.f64 55.7

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

7. Applied rewrites 55.7%

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

8. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 32.6

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

10. Applied rewrites 32.6%

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

Developer Target 1: 99.5% 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 2024313 
(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))))