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

Percentage Accurate: 99.4% → 99.8%

Time: 20.4s

Alternatives: 14

Speedup: 1.1×

• 44.1% 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 99.8%

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

   1. associate-*l* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. --lowering--.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. exp-sqrt N/A

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

   6. sqrt-unprod N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. exp-lowering-exp.f64 N/A

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

   13. *-lowering-*.f64 99.8

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

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: 93.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;e^{\frac{t \cdot t}{2}} \leq 1.0004:\\ \;\;\;\;\left(t\_1 \cdot \sqrt{2 \cdot z}\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t \cdot 0.125, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(t\_1 \cdot \sqrt{2}\right) \cdot \left(0.125 \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot t\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (exp (/ (* t t) 2.0)) 1.0004)
     (* (* t_1 (sqrt (* 2.0 z))) (fma (* t t) (fma t (* t 0.125) 0.5) 1.0))
     (* (sqrt z) (* (* t_1 (sqrt 2.0)) (* 0.125 (* (* t t) (* t t))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (exp(((t * t) / 2.0)) <= 1.0004) {
		tmp = (t_1 * sqrt((2.0 * z))) * fma((t * t), fma(t, (t * 0.125), 0.5), 1.0);
	} else {
		tmp = sqrt(z) * ((t_1 * sqrt(2.0)) * (0.125 * ((t * t) * (t * t))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (exp(Float64(Float64(t * t) / 2.0)) <= 1.0004)
		tmp = Float64(Float64(t_1 * sqrt(Float64(2.0 * z))) * fma(Float64(t * t), fma(t, Float64(t * 0.125), 0.5), 1.0));
	else
		tmp = Float64(sqrt(z) * Float64(Float64(t_1 * sqrt(2.0)) * Float64(0.125 * Float64(Float64(t * t) * Float64(t * t)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 1.0004], N[(N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * N[(t * N[(t * 0.125), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[z], $MachinePrecision] * N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(N[(t * t), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (/.f64 (*.f64 t t) #s(literal 2 binary64))) < 1.0004

   1. Initial program 99.6%

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

   3. Taylor expanded in t around 0

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

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 99.3

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

   5. Simplified 99.3%

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

   if 1.0004 < (exp.f64 (/.f64 (*.f64 t t) #s(literal 2 binary64)))

   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

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

   5. Simplified 88.5%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 89.2

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

   8. Simplified 89.2%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{t \cdot t}{2}} \leq 1.0004:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t \cdot 0.125, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right) \cdot \left(0.125 \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot t\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. 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. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 95.4

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

5. Simplified 95.4%

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

6. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. associate-*r* N/A

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

8. Simplified 96.7%

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

Alternative 4: 90.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+128}:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(t \cdot t, z \cdot \mathsf{fma}\left(t, t, 2\right), 2 \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(t\_1 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(0.5, t \cdot t, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t_1 5e+128)
     (* t_1 (sqrt (fma (* t t) (* z (fma t t 2.0)) (* 2.0 z))))
     (* (sqrt z) (* (* t_1 (sqrt 2.0)) (fma 0.5 (* t t) 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t_1 <= 5e+128) {
		tmp = t_1 * sqrt(fma((t * t), (z * fma(t, t, 2.0)), (2.0 * z)));
	} else {
		tmp = sqrt(z) * ((t_1 * sqrt(2.0)) * fma(0.5, (t * t), 1.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t_1 <= 5e+128)
		tmp = Float64(t_1 * sqrt(fma(Float64(t * t), Float64(z * fma(t, t, 2.0)), Float64(2.0 * z))));
	else
		tmp = Float64(sqrt(z) * Float64(Float64(t_1 * sqrt(2.0)) * fma(0.5, Float64(t * t), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+128], N[(t$95$1 * N[Sqrt[N[(N[(t * t), $MachinePrecision] * N[(z * N[(t * t + 2.0), $MachinePrecision]), $MachinePrecision] + N[(2.0 * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[z], $MachinePrecision] * N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x #s(literal 1/2 binary64)) y) < 5e128

   1. Initial program 99.8%

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

      1. associate-*l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. exp-sqrt N/A

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

      6. sqrt-unprod N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-lowering-exp.f64 N/A

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

      13. *-lowering-*.f64 99.8

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

   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. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. distribute-rgt-out N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 90.6

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

   7. Simplified 90.6%

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

   if 5e128 < (-.f64 (*.f64 x #s(literal 1/2 binary64)) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

   5. Simplified 92.7%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 - y \leq 5 \cdot 10^{+128}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\mathsf{fma}\left(t \cdot t, z \cdot \mathsf{fma}\left(t, t, 2\right), 2 \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(0.5, t \cdot t, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. 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. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 95.4

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

5. Simplified 95.4%

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

   1. associate-*l* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. --lowering--.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 N/A

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

7. Applied egg-rr 95.8%

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

8. Final simplification 95.8%

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

Alternative 6: 95.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. 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. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 95.4

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

5. Simplified 95.4%

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

6. Final simplification 95.4%

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

Alternative 7: 76.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \cdot t \leq 2.45 \cdot 10^{+14}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{elif}\;t \cdot t \leq 3.9 \cdot 10^{+167}:\\ \;\;\;\;t\_1 \cdot \frac{y \cdot \left(-y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\mathsf{fma}\left(0.5, t \cdot t, 1\right) \cdot \left(-t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (<= (* t t) 2.45e+14)
     (* (- (* x 0.5) y) t_1)
     (if (<= (* t t) 3.9e+167)
       (* t_1 (/ (* y (- y)) y))
       (* y (* (fma 0.5 (* t t) 1.0) (- t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if ((t * t) <= 2.45e+14) {
		tmp = ((x * 0.5) - y) * t_1;
	} else if ((t * t) <= 3.9e+167) {
		tmp = t_1 * ((y * -y) / y);
	} else {
		tmp = y * (fma(0.5, (t * t), 1.0) * -t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (Float64(t * t) <= 2.45e+14)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	elseif (Float64(t * t) <= 3.9e+167)
		tmp = Float64(t_1 * Float64(Float64(y * Float64(-y)) / y));
	else
		tmp = Float64(y * Float64(fma(0.5, Float64(t * t), 1.0) * Float64(-t_1)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 2.45e+14], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 3.9e+167], N[(t$95$1 * N[(N[(y * (-y)), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(0.5 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * (-t$95$1)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 2.45e14

   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 96.6%

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

      1. *-rgt-identity N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 96.6

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

   7. Applied egg-rr 96.6%

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

   if 2.45e14 < (*.f64 t t) < 3.8999999999999998e167

   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

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

   5. Simplified 4.4%

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

   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. neg-lowering-neg.f64 3.5

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

   8. Simplified 3.5%

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. neg-lowering-neg.f64 3.5

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

   10. Applied egg-rr 3.5%

       \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot \left(-y\right)} \]
   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. /-lowering-/.f64 N/A

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

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

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

       8. *-lowering-*.f64 N/A

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

       9. neg-lowering-neg.f64 33.3

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

   12. Applied egg-rr 33.3%

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

   if 3.8999999999999998e167 < (*.f64 t t)

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

   5. Simplified 83.3%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

      18. *-lowering-*.f64 57.9

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

   8. Simplified 57.9%

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. sqrt-prod N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. neg-lowering-neg.f64 57.9

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

   10. Applied egg-rr 57.9%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 2.45 \cdot 10^{+14}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \cdot t \leq 3.9 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \frac{y \cdot \left(-y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\mathsf{fma}\left(0.5, t \cdot t, 1\right) \cdot \left(-\sqrt{2 \cdot z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 92.3

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

5. Simplified 92.3%

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

6. Final simplification 92.3%

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

Alternative 9: 90.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. --lowering--.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. exp-sqrt N/A

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

   6. sqrt-unprod N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. exp-lowering-exp.f64 N/A

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

   13. *-lowering-*.f64 99.8

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

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. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. distribute-rgt-out N/A

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

   6. *-lowering-*.f64 N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. *-lowering-*.f64 88.6

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

7. Simplified 88.6%

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

Alternative 10: 75.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \cdot t \leq 0.00049:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\left(-y\right) \cdot \mathsf{fma}\left(0.5, t \cdot t, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (<= (* t t) 0.00049)
     (* (- (* x 0.5) y) t_1)
     (* t_1 (* (- y) (fma 0.5 (* t t) 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if ((t * t) <= 0.00049) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = t_1 * (-y * fma(0.5, (t * t), 1.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (Float64(t * t) <= 0.00049)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(t_1 * Float64(Float64(-y) * fma(0.5, Float64(t * t), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 0.00049], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * N[((-y) * N[(0.5 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 4.8999999999999998e-4

   1. Initial program 99.6%

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

   3. Taylor expanded in t around 0

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

   5. Simplified 98.5%

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

      1. *-rgt-identity N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 98.5

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

   7. Applied egg-rr 98.5%

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

   if 4.8999999999999998e-4 < (*.f64 t t)

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

   5. Simplified 70.3%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

      18. *-lowering-*.f64 45.9

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

   8. Simplified 45.9%

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

      1. neg-lowering-neg.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. sqrt-prod N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-lowering-*.f64 45.9

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

   10. Applied egg-rr 45.9%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 0.00049:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(\left(-y\right) \cdot \mathsf{fma}\left(0.5, t \cdot t, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. --lowering--.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. exp-sqrt N/A

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

   6. sqrt-unprod N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. exp-lowering-exp.f64 N/A

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

   13. *-lowering-*.f64 99.8

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

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. Taylor expanded in t around 0

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

   1. distribute-lft-out N/A

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

   2. *-lowering-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. *-lowering-*.f64 84.1

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

7. Simplified 84.1%

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

Alternative 12: 41.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ t_2 := \left(-y\right) \cdot t\_1\\ \mathbf{if}\;y \leq -1.65 \cdot 10^{-131}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-86}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(x, 0.5, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))) (t_2 (* (- y) t_1)))
   (if (<= y -1.65e-131) t_2 (if (<= y 1.15e-86) (* t_1 (fma x 0.5 y)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double t_2 = -y * t_1;
	double tmp;
	if (y <= -1.65e-131) {
		tmp = t_2;
	} else if (y <= 1.15e-86) {
		tmp = t_1 * fma(x, 0.5, y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	t_2 = Float64(Float64(-y) * t_1)
	tmp = 0.0
	if (y <= -1.65e-131)
		tmp = t_2;
	elseif (y <= 1.15e-86)
		tmp = Float64(t_1 * fma(x, 0.5, y));
	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[((-y) * t$95$1), $MachinePrecision]}, If[LessEqual[y, -1.65e-131], t$95$2, If[LessEqual[y, 1.15e-86], N[(t$95$1 * N[(x * 0.5 + y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6500000000000001e-131 or 1.14999999999999998e-86 < y

   1. Initial program 99.8%

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

   3. 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 57.4%

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

   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. neg-lowering-neg.f64 46.9

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

   8. Simplified 46.9%

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. neg-lowering-neg.f64 46.9

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

   10. Applied egg-rr 46.9%

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

   if -1.6500000000000001e-131 < y < 1.14999999999999998e-86

   1. Initial program 99.8%

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

      1. associate-*l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. exp-sqrt N/A

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

      6. sqrt-unprod N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-lowering-exp.f64 N/A

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

      13. *-lowering-*.f64 99.8

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

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 49.4

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

   7. Simplified 49.4%

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

   9. Applied egg-rr 46.7%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-131}:\\ \;\;\;\;\left(-y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(x, 0.5, y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \sqrt{2 \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.6% 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 99.8%

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

3. Taylor expanded in t around 0

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

5. Simplified 54.5%

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

   1. *-rgt-identity N/A

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

   2. *-lowering-*.f64 N/A

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

   3. --lowering--.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 54.5

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

7. Applied egg-rr 54.5%

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

Alternative 14: 30.5% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

5. Simplified 54.5%

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

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. neg-lowering-neg.f64 32.2

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

8. Simplified 32.2%

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

   1. *-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 N/A

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

   7. neg-lowering-neg.f64 32.2

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

10. Applied egg-rr 32.2%

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

11. Final simplification 32.2%

    \[\leadsto \left(-y\right) \cdot \sqrt{2 \cdot z} \]
12. 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 2024199 
(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))))