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

Percentage Accurate: 99.5% → 99.8%

Time: 10.1s

Alternatives: 10

Speedup: 0.6×

• 43.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. 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. lift-exp.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. exp-sqrt N/A

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

   12. sqrt-unprod N/A

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

   13. lower-sqrt.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. exp-prod N/A

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

   18. lower-pow.f64 N/A

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

   19. lower-exp.f64 99.4

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

4. Applied rewrites 99.4%

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

Alternative 2: 74.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. 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. lift-exp.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. exp-sqrt N/A

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

   12. sqrt-unprod N/A

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

   13. lower-sqrt.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. exp-prod N/A

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

   18. lower-pow.f64 N/A

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

   19. lower-exp.f64 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in t around 0

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

   1. lower-+.f64 75.2

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

7. Applied rewrites 75.2%

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

Alternative 3: 94.9% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (- (* x 0.5) y) (sqrt (* z 2.0)))
  (fma
   (* (fma (fma 0.020833333333333332 (* t t) 0.125) (* 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(fma(0.020833333333333332, (t * t), 0.125), (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(Float64(fma(fma(0.020833333333333332, Float64(t * t), 0.125), Float64(t * t), 0.5) * 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[(N[(0.020833333333333332 * N[(t * t), $MachinePrecision] + 0.125), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * t), $MachinePrecision] * t + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

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

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \]
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. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 94.9

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

5. Applied rewrites 94.9%

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

Alternative 4: 88.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 2 \leq 4 \cdot 10^{+166}:\\ \;\;\;\;\left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot \left(x \cdot 0.5 - y\right)\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(t \cdot t, z, z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 2.0) 4e+166)
   (* (* (fma (* t t) 0.5 1.0) (- (* x 0.5) y)) (sqrt (* 2.0 z)))
   (* (- (* 0.5 x) y) (sqrt (* 2.0 (fma (* t t) z z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 2.0) <= 4e+166) {
		tmp = (fma((t * t), 0.5, 1.0) * ((x * 0.5) - y)) * sqrt((2.0 * z));
	} else {
		tmp = ((0.5 * x) - y) * sqrt((2.0 * fma((t * t), z, z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 2.0) <= 4e+166)
		tmp = Float64(Float64(fma(Float64(t * t), 0.5, 1.0) * Float64(Float64(x * 0.5) - y)) * sqrt(Float64(2.0 * z)));
	else
		tmp = Float64(Float64(Float64(0.5 * x) - y) * sqrt(Float64(2.0 * fma(Float64(t * t), z, z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 2.0), $MachinePrecision], 4e+166], N[(N[(N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[(t * t), $MachinePrecision] * z + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 2 binary64)) < 3.99999999999999976e166

   1. Initial program 98.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
   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 85.4

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

   5. Applied rewrites 85.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 88.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 88.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 88.1

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

   7. Applied rewrites 88.1%

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

   if 3.99999999999999976e166 < (*.f64 z #s(literal 2 binary64))

   1. Initial program 99.9%

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

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

      10. lift-/.f64 N/A

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

      11. exp-sqrt N/A

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

      12. sqrt-unprod N/A

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

      13. lower-sqrt.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. exp-prod N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-exp.f64 99.9

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

   4. Applied rewrites 99.9%

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

   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-in 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. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 94.5

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

   7. Applied rewrites 94.5%

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

Alternative 5: 93.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right)\right)} \]
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 92.7

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

5. Applied rewrites 92.7%

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

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 93.5

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

   7. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(t \cdot t, \frac{1}{8}, \frac{1}{2}\right)\right)\right), t \cdot t, 1\right) \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot \sqrt{\color{blue}{z \cdot 2}} \]

   8. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(t \cdot t, \frac{1}{8}, \frac{1}{2}\right)\right)\right), t \cdot t, 1\right) \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot \sqrt{\color{blue}{2 \cdot z}} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(t \cdot t, \frac{1}{8}, \frac{1}{2}\right)\right)\right), t \cdot t, 1\right) \cdot \left(x \cdot \frac{1}{2} - y\right)\right) \cdot \sqrt{\color{blue}{2 \cdot z}} \]

7. Applied rewrites 93.5%

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

Alternative 6: 89.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. 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. lift-exp.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. exp-sqrt N/A

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

   12. sqrt-unprod N/A

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

   13. lower-sqrt.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. exp-prod N/A

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

   18. lower-pow.f64 N/A

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

   19. lower-exp.f64 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in t around 0

   \[\leadsto \left(\frac{1}{2} \cdot x - y\right) \cdot \sqrt{\color{blue}{2 \cdot z + {t}^{2} \cdot \left(2 \cdot z + {t}^{2} \cdot z\right)}} \]
6. 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. lower-fma.f64 N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 91.3

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

7. Applied rewrites 91.3%

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

Alternative 7: 73.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot t \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \sqrt{\mathsf{fma}\left(t \cdot t, z, z\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* t t) 2e+14)
   (* (- (* 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) <= 2e+14) {
		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) <= 2e+14)
		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], 2e+14], 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) < 2e14

   1. Initial program 98.9%

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

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

      10. lift-/.f64 N/A

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

      11. exp-sqrt N/A

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

      12. sqrt-unprod N/A

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

      13. lower-sqrt.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. exp-prod N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-exp.f64 98.9

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

   4. Applied rewrites 98.9%

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

   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 97.2

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

   7. Applied rewrites 97.2%

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

   if 2e14 < (*.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. lift-exp.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. exp-sqrt N/A

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

      12. sqrt-unprod N/A

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

      13. lower-sqrt.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. exp-prod N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   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 11.2

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

   7. Applied rewrites 11.2%

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

   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 7.0

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

   10. Applied rewrites 7.0%

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

   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 57.6

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

   13. Applied rewrites 57.6%

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

Alternative 8: 83.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. 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. lift-exp.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. exp-sqrt N/A

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

   12. sqrt-unprod N/A

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

   13. lower-sqrt.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. exp-prod N/A

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

   18. lower-pow.f64 N/A

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

   19. lower-exp.f64 99.4

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

4. Applied rewrites 99.4%

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

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-in 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. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 84.9

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

7. Applied rewrites 84.9%

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

Alternative 9: 56.5% 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 99.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.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. lift-exp.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. exp-sqrt N/A

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

   12. sqrt-unprod N/A

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

   13. lower-sqrt.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. exp-prod N/A

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

   18. lower-pow.f64 N/A

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

   19. lower-exp.f64 99.4

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

4. Applied rewrites 99.4%

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

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 53.9

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

7. Applied rewrites 53.9%

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

Alternative 10: 29.5% 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 99.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.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. lift-exp.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. exp-sqrt N/A

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

   12. sqrt-unprod N/A

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

   13. lower-sqrt.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. exp-prod N/A

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

   18. lower-pow.f64 N/A

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

   19. lower-exp.f64 99.4

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

4. Applied rewrites 99.4%

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

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 53.9

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

7. Applied rewrites 53.9%

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

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 28.9

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

10. Applied rewrites 28.9%

    \[\leadsto \color{blue}{\left(-y\right)} \cdot \sqrt{2 \cdot z} \]
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 2024320 
(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))))