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

Percentage Accurate: 99.4% → 99.8%

Time: 1.1min

Alternatives: 14

Speedup: 1.1×

• 45.1% 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, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-exp.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. exp-sqrt N/A

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

   14. sqrt-unprod N/A

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

   15. lower-sqrt.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. associate-*l* N/A

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

   19. lower-*.f64 N/A

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

   20. lower-*.f64 N/A

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

   21. lower-exp.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 83.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;e^{\frac{t \cdot t}{2}} \leq 2:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot \left(z \cdot \left(t \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (exp(((t * t) / 2.0d0)) <= 2.0d0) then
        tmp = t_1 * sqrt((2.0d0 * z))
    else
        tmp = t_1 * sqrt((2.0d0 * (z * (t * t))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (Math.exp(((t * t) / 2.0)) <= 2.0) {
		tmp = t_1 * Math.sqrt((2.0 * z));
	} else {
		tmp = t_1 * Math.sqrt((2.0 * (z * (t * t))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if math.exp(((t * t) / 2.0)) <= 2.0:
		tmp = t_1 * math.sqrt((2.0 * z))
	else:
		tmp = t_1 * math.sqrt((2.0 * (z * (t * t))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (exp(Float64(Float64(t * t) / 2.0)) <= 2.0)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	else
		tmp = Float64(t_1 * sqrt(Float64(2.0 * Float64(z * Float64(t * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (exp(((t * t) / 2.0)) <= 2.0)
		tmp = t_1 * sqrt((2.0 * z));
	else
		tmp = t_1 * sqrt((2.0 * (z * (t * t))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (/.f64 (*.f64 t t) #s(literal 2 binary64))) < 2

   1. Initial program 99.5%

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

   3. 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 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-rgt-identity 98.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.0

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

   7. Applied rewrites 98.0%

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

   if 2 < (exp.f64 (/.f64 (*.f64 t t) #s(literal 2 binary64)))

   1. Initial program 98.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. exp-sqrt N/A

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

      14. sqrt-unprod N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 74.0

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

   7. Applied rewrites 74.0%

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

   8. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 74.0

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

   10. Applied rewrites 74.0%

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

Alternative 3: 97.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \cdot t \leq 0.0005:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{2 \cdot z}\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right), 0.5\right), 1\right)\\ \mathbf{elif}\;t \cdot t \leq 10^{+94}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot e^{t \cdot t}\right)} \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot \left(z \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* t t) 0.0005)
     (*
      (* t_1 (sqrt (* 2.0 z)))
      (fma
       (* t t)
       (fma (* t t) (fma (* t t) 0.020833333333333332 0.125) 0.5)
       1.0))
     (if (<= (* t t) 1e+94)
       (* (sqrt (* 2.0 (* z (exp (* t t))))) (* x 0.5))
       (*
        t_1
        (sqrt
         (*
          2.0
          (*
           z
           (fma
            (* t t)
            (fma (* t t) (fma (* t t) 0.16666666666666666 0.5) 1.0)
            1.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((t * t) <= 0.0005) {
		tmp = (t_1 * sqrt((2.0 * z))) * fma((t * t), fma((t * t), fma((t * t), 0.020833333333333332, 0.125), 0.5), 1.0);
	} else if ((t * t) <= 1e+94) {
		tmp = sqrt((2.0 * (z * exp((t * t))))) * (x * 0.5);
	} else {
		tmp = t_1 * sqrt((2.0 * (z * fma((t * t), fma((t * t), fma((t * t), 0.16666666666666666, 0.5), 1.0), 1.0))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(t * t) <= 0.0005)
		tmp = Float64(Float64(t_1 * sqrt(Float64(2.0 * z))) * fma(Float64(t * t), fma(Float64(t * t), fma(Float64(t * t), 0.020833333333333332, 0.125), 0.5), 1.0));
	elseif (Float64(t * t) <= 1e+94)
		tmp = Float64(sqrt(Float64(2.0 * Float64(z * exp(Float64(t * t))))) * Float64(x * 0.5));
	else
		tmp = Float64(t_1 * sqrt(Float64(2.0 * Float64(z * fma(Float64(t * t), fma(Float64(t * t), fma(Float64(t * t), 0.16666666666666666, 0.5), 1.0), 1.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 0.0005], N[(N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.020833333333333332 + 0.125), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 1e+94], N[(N[Sqrt[N[(2.0 * N[(z * N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(2.0 * N[(z * N[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 5.0000000000000001e-4

   1. Initial program 99.5%

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

   3. 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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 5.0000000000000001e-4 < (*.f64 t t) < 1e94

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. exp-sqrt N/A

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

      14. sqrt-unprod N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 81.3

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

   7. Applied rewrites 81.3%

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

   if 1e94 < (*.f64 t t)

   1. Initial program 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. exp-sqrt N/A

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

      14. sqrt-unprod N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \left(z \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 0.0005:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right), 0.5\right), 1\right)\\ \mathbf{elif}\;t \cdot t \leq 10^{+94}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot e^{t \cdot t}\right)} \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \left(z \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-exp.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. exp-sqrt N/A

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

   14. sqrt-unprod N/A

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

   15. lower-sqrt.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. associate-*l* N/A

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

   19. lower-*.f64 N/A

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

   20. lower-*.f64 N/A

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

   21. lower-exp.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 95.3

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

7. Applied rewrites 95.3%

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

Alternative 5: 94.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 95.3

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

5. Applied rewrites 95.3%

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

6. Final simplification 95.3%

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

Alternative 6: 88.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;2 \cdot z \leq 5 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{z} \cdot \left(t\_1 \cdot \left(\mathsf{fma}\left(0.5, t \cdot t, 1\right) \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot \mathsf{fma}\left(z, t \cdot t, z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* 2.0 z) 5e+84)
     (* (sqrt z) (* t_1 (* (fma 0.5 (* t t) 1.0) (sqrt 2.0))))
     (* t_1 (sqrt (* 2.0 (fma z (* t t) z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((2.0 * z) <= 5e+84) {
		tmp = sqrt(z) * (t_1 * (fma(0.5, (t * t), 1.0) * sqrt(2.0)));
	} else {
		tmp = t_1 * sqrt((2.0 * fma(z, (t * t), z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(2.0 * z) <= 5e+84)
		tmp = Float64(sqrt(z) * Float64(t_1 * Float64(fma(0.5, Float64(t * t), 1.0) * sqrt(2.0))));
	else
		tmp = Float64(t_1 * sqrt(Float64(2.0 * fma(z, Float64(t * t), z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[(2.0 * z), $MachinePrecision], 5e+84], N[(N[Sqrt[z], $MachinePrecision] * N[(t$95$1 * N[(N[(0.5 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(2.0 * N[(z * N[(t * t), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 2 binary64)) < 5.0000000000000001e84

   1. Initial program 98.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 86.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 86.2

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

   7. Applied rewrites 86.2%

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

   if 5.0000000000000001e84 < (*.f64 z #s(literal 2 binary64))

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. exp-sqrt N/A

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

      14. sqrt-unprod N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-exp.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot z \leq 5 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(\mathsf{fma}\left(0.5, t \cdot t, 1\right) \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(z, t \cdot t, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \cdot t \leq 2 \cdot 10^{+55}:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot \mathsf{fma}\left(z, t \cdot t, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(t\_1 \cdot \left(0.5 \cdot \left(t \cdot \left(t \cdot \sqrt{2}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* t t) 2e+55)
     (* t_1 (sqrt (* 2.0 (fma z (* t t) z))))
     (* (sqrt z) (* t_1 (* 0.5 (* t (* t (sqrt 2.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((t * t) <= 2e+55) {
		tmp = t_1 * sqrt((2.0 * fma(z, (t * t), z)));
	} else {
		tmp = sqrt(z) * (t_1 * (0.5 * (t * (t * sqrt(2.0)))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(t * t) <= 2e+55)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * fma(z, Float64(t * t), z))));
	else
		tmp = Float64(sqrt(z) * Float64(t_1 * Float64(0.5 * Float64(t * Float64(t * sqrt(2.0))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 2e+55], N[(t$95$1 * N[Sqrt[N[(2.0 * N[(z * N[(t * t), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[z], $MachinePrecision] * N[(t$95$1 * N[(0.5 * N[(t * N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 2.00000000000000002e55

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. exp-sqrt N/A

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

      14. sqrt-unprod N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-exp.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 93.5

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

   7. Applied rewrites 93.5%

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

   if 2.00000000000000002e55 < (*.f64 t t)

   1. Initial program 98.3%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 84.2

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

   8. Applied rewrites 84.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 84.2

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

   10. Applied rewrites 84.2%

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

Alternative 8: 91.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right)\right)} \]
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. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 92.7

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

5. Applied rewrites 92.7%

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

6. Final simplification 92.7%

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

Alternative 9: 87.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot t \leq 2 \cdot 10^{+55}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(z, t \cdot t, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t \cdot t\right) \cdot \mathsf{fma}\left(x, 0.25, y \cdot -0.5\right)\right) \cdot \sqrt{2 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* t t) 2e+55)
   (* (- (* x 0.5) y) (sqrt (* 2.0 (fma z (* t t) z))))
   (* (* (* t t) (fma x 0.25 (* y -0.5))) (sqrt (* 2.0 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * t) <= 2e+55) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * fma(z, (t * t), z)));
	} else {
		tmp = ((t * t) * fma(x, 0.25, (y * -0.5))) * sqrt((2.0 * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(t * t) <= 2e+55)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * fma(z, Float64(t * t), z))));
	else
		tmp = Float64(Float64(Float64(t * t) * fma(x, 0.25, Float64(y * -0.5))) * sqrt(Float64(2.0 * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(t * t), $MachinePrecision], 2e+55], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(z * N[(t * t), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * t), $MachinePrecision] * N[(x * 0.25 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 2.00000000000000002e55

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. exp-sqrt N/A

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

      14. sqrt-unprod N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-exp.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 93.5

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

   7. Applied rewrites 93.5%

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

   if 2.00000000000000002e55 < (*.f64 t t)

   1. Initial program 98.3%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 84.2

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

   8. Applied rewrites 84.2%

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

   10. Applied rewrites 84.2%

       \[\leadsto \color{blue}{\left(\left(t \cdot t\right) \cdot \mathsf{fma}\left(x, 0.25, -0.5 \cdot y\right)\right) \cdot \sqrt{2 \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 2 \cdot 10^{+55}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(z, t \cdot t, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t \cdot t\right) \cdot \mathsf{fma}\left(x, 0.25, y \cdot -0.5\right)\right) \cdot \sqrt{2 \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot \mathsf{fma}\left(z, t \cdot t, z\right)}\\ \mathbf{if}\;t \leq 8.2 \cdot 10^{+19}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+286}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 (fma z (* t t) z)))))
   (if (<= t 8.2e+19)
     (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
     (if (<= t 4.2e+286) (* (* x 0.5) t_1) (* t_1 (- y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * fma(z, (t * t), z)));
	double tmp;
	if (t <= 8.2e+19) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	} else if (t <= 4.2e+286) {
		tmp = (x * 0.5) * t_1;
	} else {
		tmp = t_1 * -y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * fma(z, Float64(t * t), z)))
	tmp = 0.0
	if (t <= 8.2e+19)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	elseif (t <= 4.2e+286)
		tmp = Float64(Float64(x * 0.5) * t_1);
	else
		tmp = Float64(t_1 * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(z * N[(t * t), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 8.2e+19], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e+286], N[(N[(x * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * (-y)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 8.2e19

   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 68.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-rgt-identity 68.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 68.1

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

   7. Applied rewrites 68.1%

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

   if 8.2e19 < t < 4.2000000000000002e286

   1. Initial program 96.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. exp-sqrt N/A

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

      14. sqrt-unprod N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 75.3

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

   7. Applied rewrites 75.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 63.1

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

   10. Applied rewrites 63.1%

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

   if 4.2000000000000002e286 < t

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. exp-sqrt N/A

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

      14. sqrt-unprod N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 88.9

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

   10. Applied rewrites 88.9%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{+19}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+286}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(z, t \cdot t, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(z, t \cdot t, z\right)} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{+56}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(z, t \cdot t, z\right)} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 3.7e+56)
   (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
   (* (sqrt (* 2.0 (fma z (* t t) z))) (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3.7e+56) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	} else {
		tmp = sqrt((2.0 * fma(z, (t * t), z))) * -y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 3.7e+56)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	else
		tmp = Float64(sqrt(Float64(2.0 * fma(z, Float64(t * t), z))) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 3.7e+56], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(z * N[(t * t), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.69999999999999997e56

   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. 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 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-rgt-identity 67.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 67.1

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

   7. Applied rewrites 67.1%

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

   if 3.69999999999999997e56 < t

   1. Initial program 98.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. exp-sqrt N/A

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

      14. sqrt-unprod N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.1

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

   7. Applied rewrites 82.1%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 66.7

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

   10. Applied rewrites 66.7%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{+56}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(z, t \cdot t, z\right)} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-exp.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. exp-sqrt N/A

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

   14. sqrt-unprod N/A

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

   15. lower-sqrt.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. associate-*l* N/A

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

   19. lower-*.f64 N/A

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

   20. lower-*.f64 N/A

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

   21. lower-exp.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 86.2

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

7. Applied rewrites 86.2%

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

Alternative 13: 57.2% accurate, 5.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * 0.5) - 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 = ((x * 0.5d0) - y) * sqrt((2.0d0 * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $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. 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 56.4%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-rgt-identity 56.4

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 56.4

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

7. Applied rewrites 56.4%

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

Alternative 14: 29.3% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Applied rewrites 56.4%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 32.9

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

8. Applied rewrites 32.9%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-rgt-identity 32.9

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 32.9

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 32.9

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

10. Applied rewrites 32.9%

    \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot \left(-y\right)} \]
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 2024214 
(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))))