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

Percentage Accurate: 99.4% → 99.8%

Time: 14.5s

Alternatives: 10

Speedup: 0.6×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% 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: 74.8% 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 74.0

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

7. Applied rewrites 74.0%

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

Alternative 3: 95.9% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (*
   (fma (fma (fma (* 0.020833333333333332 t) t 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((0.020833333333333332 * t), t, 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(0.020833333333333332 * t), t, 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[(0.020833333333333332 * t), $MachinePrecision] * t + 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 92.4

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

5. Applied rewrites 92.4%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)} \]
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 94.2%

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

9. Applied rewrites 94.2%

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

Alternative 4: 95.7% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (*
   (fma (fma (* 0.020833333333333332 (* t t)) (* 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((0.020833333333333332 * (t * t)), (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(Float64(0.020833333333333332 * Float64(t * t)), 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[(0.020833333333333332 * N[(t * t), $MachinePrecision]), $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 92.4

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

5. Applied rewrites 92.4%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)} \]
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 94.2%

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

8. Taylor expanded in t around inf

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

10. Applied rewrites 94.2%

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

Alternative 5: 94.0% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (fma (fma 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(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(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[(0.125 * 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 92.4

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

5. Applied rewrites 92.4%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)} \]
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 94.2%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 91.6%

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

Alternative 6: 73.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 2 \cdot 10^{+131}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot t\_1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot \left(x \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (* t t) 2e+131)
     (* (* (- (* x 0.5) y) t_1) 1.0)
     (* (* t_1 (* x 0.5)) (fma (* t t) 0.5 1.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 2e+131) {
		tmp = (((x * 0.5) - y) * t_1) * 1.0;
	} else {
		tmp = (t_1 * (x * 0.5)) * fma((t * t), 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (Float64(t * t) <= 2e+131)
		tmp = Float64(Float64(Float64(Float64(x * 0.5) - y) * t_1) * 1.0);
	else
		tmp = Float64(Float64(t_1 * Float64(x * 0.5)) * fma(Float64(t * t), 0.5, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 2e+131], N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(t$95$1 * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 1.9999999999999998e131

   1. Initial program 99.1%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 80.6%

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

   if 1.9999999999999998e131 < (*.f64 t t)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.6

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

   8. Applied rewrites 63.6%

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

   10. Applied rewrites 63.6%

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

Alternative 7: 74.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 10^{+91}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot t\_1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot t\_1\right) \cdot \mathsf{fma}\left(0.5 \cdot t, t, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (* t t) 1e+91)
     (* (* (- (* x 0.5) y) t_1) 1.0)
     (* (* (- y) t_1) (fma (* 0.5 t) t 1.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 1e+91) {
		tmp = (((x * 0.5) - y) * t_1) * 1.0;
	} else {
		tmp = (-y * t_1) * fma((0.5 * t), t, 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (Float64(t * t) <= 1e+91)
		tmp = Float64(Float64(Float64(Float64(x * 0.5) - y) * t_1) * 1.0);
	else
		tmp = Float64(Float64(Float64(-y) * t_1) * fma(Float64(0.5 * t), t, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 1e+91], N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[((-y) * t$95$1), $MachinePrecision] * N[(N[(0.5 * t), $MachinePrecision] * t + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 1.00000000000000008e91

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 84.4%

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

   if 1.00000000000000008e91 < (*.f64 t t)

   1. Initial program 99.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 6.8

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

   8. Applied rewrites 6.8%

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

   9. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 52.9

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

   11. Applied rewrites 52.9%

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

   13. Applied rewrites 52.9%

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

Alternative 8: 87.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(t * t), $MachinePrecision] * 0.5 + 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 92.4

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

5. Applied rewrites 92.4%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)} \]
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 94.2%

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

8. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 85.2

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

10. Applied rewrites 85.2%

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

Alternative 9: 56.9% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

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))) * 1.0d0
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))) * 1.0;
}

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * 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] * 1.0), $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

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

5. Applied rewrites 56.7%

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

Alternative 10: 29.1% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

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 = (sqrt((z * 2.0d0)) * -y) * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Applied rewrites 56.7%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

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

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

   5. lower-*.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-sqrt.f64 30.7

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

8. Applied rewrites 30.7%

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

10. Applied rewrites 30.8%

    \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(-y\right)\right) \cdot 1} \]
11. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024324 
(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))))