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

Percentage Accurate: 99.4% → 99.8%

Time: 39.8s

Alternatives: 15

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 egg-rr 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: 65.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{t \cdot t}{2}} \leq 2.1 \cdot 10^{+70}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{z}\right) \cdot \left(y \cdot \left(y \cdot \left(-y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (exp (/ (* t t) 2.0)) 2.1e+70)
   (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
   (* (* (sqrt 2.0) (sqrt z)) (* y (* y (- y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (exp(((t * t) / 2.0)) <= 2.1e+70) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	} else {
		tmp = (sqrt(2.0) * sqrt(z)) * (y * (y * -y));
	}
	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) :: tmp
    if (exp(((t * t) / 2.0d0)) <= 2.1d+70) then
        tmp = ((x * 0.5d0) - y) * sqrt((2.0d0 * z))
    else
        tmp = (sqrt(2.0d0) * sqrt(z)) * (y * (y * -y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (Math.exp(((t * t) / 2.0)) <= 2.1e+70) {
		tmp = ((x * 0.5) - y) * Math.sqrt((2.0 * z));
	} else {
		tmp = (Math.sqrt(2.0) * Math.sqrt(z)) * (y * (y * -y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if math.exp(((t * t) / 2.0)) <= 2.1e+70:
		tmp = ((x * 0.5) - y) * math.sqrt((2.0 * z))
	else:
		tmp = (math.sqrt(2.0) * math.sqrt(z)) * (y * (y * -y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (exp(Float64(Float64(t * t) / 2.0)) <= 2.1e+70)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	else
		tmp = Float64(Float64(sqrt(2.0) * sqrt(z)) * Float64(y * Float64(y * Float64(-y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (exp(((t * t) / 2.0)) <= 2.1e+70)
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	else
		tmp = (sqrt(2.0) * sqrt(z)) * (y * (y * -y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.1e+70], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * (-y)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (/.f64 (*.f64 t t) #s(literal 2 binary64))) < 2.10000000000000008e70

   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. Simplified 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 egg-rr 98.0%

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

   if 2.10000000000000008e70 < (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. Taylor expanded in t around 0

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

   5. Simplified 17.3%

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

   7. Applied egg-rr 10.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.125, x \cdot \left(x \cdot x\right), -y \cdot \left(y \cdot y\right)\right) \cdot \sqrt{2 \cdot z}}{\mathsf{fma}\left(x, x \cdot 0.25, y \cdot \mathsf{fma}\left(x, 0.5, y\right)\right)}} \]

   8. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left({y}^{3} \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{y}^{3} \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left({y}^{3} \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)}\right) \]

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

         \[\leadsto \color{blue}{{y}^{3} \cdot \left(\mathsf{neg}\left(\sqrt{z} \cdot \sqrt{2}\right)\right)} \]

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. mul-1-neg N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-sqrt.f64 32.5

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

   10. Simplified 32.5%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{t \cdot t}{2}} \leq 2.1 \cdot 10^{+70}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{z}\right) \cdot \left(y \cdot \left(y \cdot \left(-y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ t_2 := \left(\left(x \cdot 0.5 - y\right) \cdot t\_1\right) \cdot \mathsf{fma}\left(0.5, t \cdot t, 1\right)\\ \mathbf{if}\;t \cdot t \leq 5 \cdot 10^{+34}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \cdot t \leq 10^{+297}:\\ \;\;\;\;t\_1 \cdot \left(\left(x \cdot 0.5\right) \cdot \mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot 0.125, 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z)))
        (t_2 (* (* (- (* x 0.5) y) t_1) (fma 0.5 (* t t) 1.0))))
   (if (<= (* t t) 5e+34)
     t_2
     (if (<= (* t t) 1e+297)
       (* t_1 (* (* x 0.5) (fma t (* t (fma t (* t 0.125) 0.5)) 1.0)))
       t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double t_2 = (((x * 0.5) - y) * t_1) * fma(0.5, (t * t), 1.0);
	double tmp;
	if ((t * t) <= 5e+34) {
		tmp = t_2;
	} else if ((t * t) <= 1e+297) {
		tmp = t_1 * ((x * 0.5) * fma(t, (t * fma(t, (t * 0.125), 0.5)), 1.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	t_2 = Float64(Float64(Float64(Float64(x * 0.5) - y) * t_1) * fma(0.5, Float64(t * t), 1.0))
	tmp = 0.0
	if (Float64(t * t) <= 5e+34)
		tmp = t_2;
	elseif (Float64(t * t) <= 1e+297)
		tmp = Float64(t_1 * Float64(Float64(x * 0.5) * fma(t, Float64(t * fma(t, Float64(t * 0.125), 0.5)), 1.0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(0.5 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 5e+34], t$95$2, If[LessEqual[N[(t * t), $MachinePrecision], 1e+297], N[(t$95$1 * N[(N[(x * 0.5), $MachinePrecision] * N[(t * N[(t * N[(t * N[(t * 0.125), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 4.9999999999999998e34 or 1e297 < (*.f64 t t)

   1. Initial program 99.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 96.3

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

   5. Simplified 96.3%

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

   if 4.9999999999999998e34 < (*.f64 t t) < 1e297

   1. Initial program 97.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. 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. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 72.8

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

   5. Simplified 72.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 50.2

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

   8. Simplified 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   10. Applied egg-rr 63.0%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 5 \cdot 10^{+34}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot \mathsf{fma}\left(0.5, t \cdot t, 1\right)\\ \mathbf{elif}\;t \cdot t \leq 10^{+297}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(\left(x \cdot 0.5\right) \cdot \mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot 0.125, 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot \mathsf{fma}\left(0.5, t \cdot t, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ t_2 := \left(\left(x \cdot 0.5 - y\right) \cdot t\_1\right) \cdot \mathsf{fma}\left(0.5, t \cdot t, 1\right)\\ \mathbf{if}\;t \cdot t \leq 2.5 \cdot 10^{+112}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \cdot t \leq 10^{+297}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \left(t\_1 \cdot \mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot 0.125, 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z)))
        (t_2 (* (* (- (* x 0.5) y) t_1) (fma 0.5 (* t t) 1.0))))
   (if (<= (* t t) 2.5e+112)
     t_2
     (if (<= (* t t) 1e+297)
       (* (* x 0.5) (* t_1 (fma t (* t (fma t (* t 0.125) 0.5)) 1.0)))
       t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double t_2 = (((x * 0.5) - y) * t_1) * fma(0.5, (t * t), 1.0);
	double tmp;
	if ((t * t) <= 2.5e+112) {
		tmp = t_2;
	} else if ((t * t) <= 1e+297) {
		tmp = (x * 0.5) * (t_1 * fma(t, (t * fma(t, (t * 0.125), 0.5)), 1.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	t_2 = Float64(Float64(Float64(Float64(x * 0.5) - y) * t_1) * fma(0.5, Float64(t * t), 1.0))
	tmp = 0.0
	if (Float64(t * t) <= 2.5e+112)
		tmp = t_2;
	elseif (Float64(t * t) <= 1e+297)
		tmp = Float64(Float64(x * 0.5) * Float64(t_1 * fma(t, Float64(t * fma(t, Float64(t * 0.125), 0.5)), 1.0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(0.5 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 2.5e+112], t$95$2, If[LessEqual[N[(t * t), $MachinePrecision], 1e+297], N[(N[(x * 0.5), $MachinePrecision] * N[(t$95$1 * N[(t * N[(t * N[(t * N[(t * 0.125), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 2.5e112 or 1e297 < (*.f64 t t)

   1. Initial program 98.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 92.2

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

   5. Simplified 92.2%

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

   if 2.5e112 < (*.f64 t t) < 1e297

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. 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. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 85.5

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

   5. Simplified 85.5%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 63.2

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

   8. Simplified 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   10. Applied egg-rr 75.5%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 2.5 \cdot 10^{+112}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot \mathsf{fma}\left(0.5, t \cdot t, 1\right)\\ \mathbf{elif}\;t \cdot t \leq 10^{+297}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \left(\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(t, t \cdot 0.125, 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot \mathsf{fma}\left(0.5, t \cdot t, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ t_2 := \left(\left(x \cdot 0.5 - y\right) \cdot t\_1\right) \cdot \mathsf{fma}\left(0.5, t \cdot t, 1\right)\\ \mathbf{if}\;t \cdot t \leq 5 \cdot 10^{+55}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \cdot t \leq 10^{+297}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.125, 0.5\right), 1\right) \cdot \left(\left(-y\right) \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z)))
        (t_2 (* (* (- (* x 0.5) y) t_1) (fma 0.5 (* t t) 1.0))))
   (if (<= (* t t) 5e+55)
     t_2
     (if (<= (* t t) 1e+297)
       (* (fma (* t t) (fma (* t t) 0.125 0.5) 1.0) (* (- y) t_1))
       t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double t_2 = (((x * 0.5) - y) * t_1) * fma(0.5, (t * t), 1.0);
	double tmp;
	if ((t * t) <= 5e+55) {
		tmp = t_2;
	} else if ((t * t) <= 1e+297) {
		tmp = fma((t * t), fma((t * t), 0.125, 0.5), 1.0) * (-y * t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	t_2 = Float64(Float64(Float64(Float64(x * 0.5) - y) * t_1) * fma(0.5, Float64(t * t), 1.0))
	tmp = 0.0
	if (Float64(t * t) <= 5e+55)
		tmp = t_2;
	elseif (Float64(t * t) <= 1e+297)
		tmp = Float64(fma(Float64(t * t), fma(Float64(t * t), 0.125, 0.5), 1.0) * Float64(Float64(-y) * t_1));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(0.5 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 5e+55], t$95$2, If[LessEqual[N[(t * t), $MachinePrecision], 1e+297], N[(N[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.125 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-y) * t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 5.00000000000000046e55 or 1e297 < (*.f64 t t)

   1. Initial program 99.3%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 94.6

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

   5. Simplified 94.6%

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

   if 5.00000000000000046e55 < (*.f64 t t) < 1e297

   1. Initial program 97.4%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. 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. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 78.5

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

   5. Simplified 78.5%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 62.7

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

   8. Simplified 62.7%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 5 \cdot 10^{+55}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot \mathsf{fma}\left(0.5, t \cdot t, 1\right)\\ \mathbf{elif}\;t \cdot t \leq 10^{+297}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, 0.125, 0.5\right), 1\right) \cdot \left(\left(-y\right) \cdot \sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot \mathsf{fma}\left(0.5, t \cdot t, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.8% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (fma 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 fma(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(fma(x, 0.5, Float64(-y)) * Float64(sqrt(Float64(2.0 * z)) * fma(Float64(t * t), fma(t, Float64(t * fma(Float64(t * t), 0.020833333333333332, 0.125)), 0.5), 1.0)))
end

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 95.3

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

5. Simplified 95.3%

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

7. Applied egg-rr 96.1%

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

Alternative 7: 94.3% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (- (* x 0.5) y) (sqrt (* 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(t, Float64(t * fma(Float64(t * t), fma(Float64(t * t), 0.020833333333333332, 0.125), 0.5)), 1.0))
end

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 95.3

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

5. Simplified 95.3%

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

6. Final simplification 95.3%

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

Alternative 8: 71.3% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
  (fma (* t t) (fma (* t t) (* 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), (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(Float64(t * 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[(N[(t * t), $MachinePrecision] * 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. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 92.7

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

5. Simplified 92.7%

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

6. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. unpow3 N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 70.0

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

8. Simplified 70.0%

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

9. Final simplification 70.0%

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

Alternative 9: 93.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. 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. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 92.7

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

5. Simplified 92.7%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-rgt-identity N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. lift-fma.f64 N/A

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

   11. *-rgt-identity N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r* N/A

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

7. Applied egg-rr 94.5%

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

8. Final simplification 94.5%

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

Alternative 10: 91.8% accurate, 2.7× speedup

Math

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

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(Float64(t * t), 0.125, 0.5), 1.0))
end

Wolfram

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

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 92.7

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

5. Simplified 92.7%

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

6. Final simplification 92.7%

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

Alternative 11: 58.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \leq 0.02:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;-t\_1 \cdot \frac{y \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (<= t 0.02) (* (- (* x 0.5) y) t_1) (- (* t_1 (/ (* y y) y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if (t <= 0.02) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = -(t_1 * ((y * y) / y));
	}
	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 = sqrt((2.0d0 * z))
    if (t <= 0.02d0) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = -(t_1 * ((y * y) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((2.0 * z));
	double tmp;
	if (t <= 0.02) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = -(t_1 * ((y * y) / y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	tmp = 0
	if t <= 0.02:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = -(t_1 * ((y * y) / y))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (t <= 0.02)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(-Float64(t_1 * Float64(Float64(y * y) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	tmp = 0.0;
	if (t <= 0.02)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = -(t_1 * ((y * y) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 0.02], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], (-N[(t$95$1 * N[(N[(y * y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if t < 0.0200000000000000004

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0

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

   5. Simplified 69.7%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{1} \]
   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 69.7

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

      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 69.7

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

   7. Applied egg-rr 69.7%

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

   if 0.0200000000000000004 < t

   1. Initial program 96.9%

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

   3. Taylor expanded in t around 0

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

   5. Simplified 16.5%

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

   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 7.5

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

   8. Simplified 7.5%

      \[\leadsto \left(\color{blue}{\left(-y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]
   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 7.5

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

      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. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 7.5

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

   10. Applied egg-rr 7.5%

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

       1. neg-sub0 N/A

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

       2. flip-- N/A

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

       3. metadata-eval N/A

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

       4. neg-sub0 N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \frac{\color{blue}{\mathsf{neg}\left(y \cdot y\right)}}{0 + y} \]

       5. +-lft-identity N/A

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

       6. lower-/.f64 N/A

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

       7. lower-neg.f64 N/A

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

       8. lower-*.f64 26.7

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

   12. Applied egg-rr 26.7%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.02:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot z} \cdot \frac{y \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.3% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 85.8

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

5. Simplified 85.8%

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

6. Final simplification 85.8%

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in t around 0

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

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

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 32.9

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

10. Applied egg-rr 32.9%

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

11. Final simplification 32.9%

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

Alternative 15: 0.0% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[((-y) * N[Sqrt[(-z)], $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. Simplified 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. Simplified 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. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 32.9

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

10. Applied egg-rr 32.9%

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

11. Taylor expanded in z around -inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 0.0

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

13. Simplified 0.0%

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

14. Final simplification 0.0%

    \[\leadsto \left(-y\right) \cdot \sqrt{-z} \]
15. 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))))