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

Percentage Accurate: 99.5% → 99.5%

Time: 45.1s

Alternatives: 15

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * exp((0.5 * (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((z * 2.0d0))) * exp((0.5d0 * (t * t)))
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((0.5 * (t * t)));
}

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * exp((0.5 * (t * t)));
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[(0.5 * N[(t * t), $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. exp-lowering-exp.f64 N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

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

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

   5. *-lowering-*.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 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 3: 87.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ t_2 := \left(\left(x \cdot 0.5 - y\right) \cdot t\_1\right) \cdot \mathsf{fma}\left(0.5, t \cdot t, 1\right)\\ \mathbf{if}\;t \cdot t \leq 5 \cdot 10^{+58}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+261}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t \cdot 0.125, 0.5\right), 1\right) \cdot \left(\left(x \cdot 0.5\right) \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0)))
        (t_2 (* (* (- (* x 0.5) y) t_1) (fma 0.5 (* t t) 1.0))))
   (if (<= (* t t) 5e+58)
     t_2
     (if (<= (* t t) 5e+261)
       (* (fma (* t t) (fma t (* t 0.125) 0.5) 1.0) (* (* x 0.5) t_1))
       t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double t_2 = (((x * 0.5) - y) * t_1) * fma(0.5, (t * t), 1.0);
	double tmp;
	if ((t * t) <= 5e+58) {
		tmp = t_2;
	} else if ((t * t) <= 5e+261) {
		tmp = fma((t * t), fma(t, (t * 0.125), 0.5), 1.0) * ((x * 0.5) * t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	t_2 = Float64(Float64(Float64(Float64(x * 0.5) - y) * t_1) * fma(0.5, Float64(t * t), 1.0))
	tmp = 0.0
	if (Float64(t * t) <= 5e+58)
		tmp = t_2;
	elseif (Float64(t * t) <= 5e+261)
		tmp = Float64(fma(Float64(t * t), fma(t, Float64(t * 0.125), 0.5), 1.0) * Float64(Float64(x * 0.5) * t_1));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(0.5 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 5e+58], t$95$2, If[LessEqual[N[(t * t), $MachinePrecision], 5e+261], N[(N[(N[(t * t), $MachinePrecision] * N[(t * N[(t * 0.125), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 4.99999999999999986e58 or 5.0000000000000001e261 < (*.f64 t t)

   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

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 92.9

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

   5. Simplified 92.9%

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

   if 4.99999999999999986e58 < (*.f64 t t) < 5.0000000000000001e261

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. 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 80.6

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

   5. Simplified 80.6%

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

   6. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 69.0

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

   8. Simplified 69.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 5 \cdot 10^{+58}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(0.5, t \cdot t, 1\right)\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+261}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t, t \cdot 0.125, 0.5\right), 1\right) \cdot \left(\left(x \cdot 0.5\right) \cdot \sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(0.5, t \cdot t, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * N[(t * N[(t * N[(t * N[(t * 0.020833333333333332), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * 0.5 + (-y)), $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 96.0

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

5. Simplified 96.0%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

7. Applied egg-rr 97.1%

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

8. Final simplification 97.1%

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

Alternative 5: 95.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
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 96.0

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

5. Simplified 96.0%

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

Alternative 6: 95.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \]
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 96.0

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

5. Simplified 96.0%

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

6. Taylor expanded in t around inf

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

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 95.8

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

8. Simplified 95.8%

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

Alternative 7: 94.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right)\right)} \]
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 93.0

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

5. Simplified 93.0%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. *-lowering-*.f64 93.7

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

7. Applied egg-rr 93.7%

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

8. Final simplification 93.7%

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

Alternative 8: 93.0% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (- (* x 0.5) y) (sqrt (* z 2.0)))
  (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((z * 2.0))) * 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(z * 2.0))) * 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[(z * 2.0), $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 93.0

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

5. Simplified 93.0%

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

Alternative 9: 87.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) \cdot \sqrt{z}\right) + \sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
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 87.1%

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

6. Final simplification 87.1%

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

Alternative 10: 63.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 8 \cdot 10^{-51}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t\_1 \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (* t t) 8e-51)
     (* (- (* x 0.5) y) t_1)
     (* x (* t_1 (- 0.5 (/ y x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 8e-51) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = x * (t_1 * (0.5 - (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((t * t) <= 8d-51) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = x * (t_1 * (0.5d0 - (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 8e-51) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = x * (t_1 * (0.5 - (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (t * t) <= 8e-51:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = x * (t_1 * (0.5 - (y / x)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (Float64(t * t) <= 8e-51)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(x * Float64(t_1 * Float64(0.5 - Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((t * t) <= 8e-51)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = x * (t_1 * (0.5 - (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 8e-51], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(x * N[(t$95$1 * N[(0.5 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 8.0000000000000001e-51

   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 99.6%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{1} \]
   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. *-lowering-*.f64 99.6

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

   7. Applied egg-rr 99.6%

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

   if 8.0000000000000001e-51 < (*.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

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

   5. Simplified 15.0%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. unsub-neg N/A

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

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

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

      15. /-lowering-/.f64 24.3

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

   8. Simplified 24.3%

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

      1. *-rgt-identity N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

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

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

      10. /-lowering-/.f64 27.9

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

   10. Applied egg-rr 27.9%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 8 \cdot 10^{-51}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{z \cdot 2} \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 1.5 \cdot 10^{+157}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \left(t\_1 \cdot \left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (* t t) 1.5e+157)
     (* (- (* x 0.5) y) t_1)
     (* (/ y x) (* t_1 (- x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 1.5e+157) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = (y / x) * (t_1 * -x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((t * t) <= 1.5d+157) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = (y / x) * (t_1 * -x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 1.5e+157) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = (y / x) * (t_1 * -x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (t * t) <= 1.5e+157:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = (y / x) * (t_1 * -x)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (Float64(t * t) <= 1.5e+157)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(Float64(y / x) * Float64(t_1 * Float64(-x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((t * t) <= 1.5e+157)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = (y / x) * (t_1 * -x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 1.5e+157], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(y / x), $MachinePrecision] * N[(t$95$1 * (-x)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 1.50000000000000005e157

   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 80.8%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{1} \]
   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. *-lowering-*.f64 80.8

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

   7. Applied egg-rr 80.8%

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

   if 1.50000000000000005e157 < (*.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

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

   5. Simplified 9.3%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. unsub-neg N/A

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

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

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

      15. /-lowering-/.f64 19.9

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

   8. Simplified 19.9%

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 16.5

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

   10. Applied egg-rr 16.5%

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

   11. Taylor expanded in y around inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

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

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

       5. mul-1-neg N/A

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

       6. neg-lowering-neg.f64 14.5

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

   13. Simplified 14.5%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 1.5 \cdot 10^{+157}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \left(\sqrt{z \cdot 2} \cdot \left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ t_2 := 0.5 \cdot \left(x \cdot t\_1\right)\\ \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \cdot 0.5 \leq 5 \cdot 10^{+37}:\\ \;\;\;\;t\_1 \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))) (t_2 (* 0.5 (* x t_1))))
   (if (<= (* x 0.5) -5e+18) t_2 (if (<= (* x 0.5) 5e+37) (* t_1 (- y)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double t_2 = 0.5 * (x * t_1);
	double tmp;
	if ((x * 0.5) <= -5e+18) {
		tmp = t_2;
	} else if ((x * 0.5) <= 5e+37) {
		tmp = t_1 * -y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    t_2 = 0.5d0 * (x * t_1)
    if ((x * 0.5d0) <= (-5d+18)) then
        tmp = t_2
    else if ((x * 0.5d0) <= 5d+37) then
        tmp = t_1 * -y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double t_2 = 0.5 * (x * t_1);
	double tmp;
	if ((x * 0.5) <= -5e+18) {
		tmp = t_2;
	} else if ((x * 0.5) <= 5e+37) {
		tmp = t_1 * -y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	t_2 = 0.5 * (x * t_1)
	tmp = 0
	if (x * 0.5) <= -5e+18:
		tmp = t_2
	elif (x * 0.5) <= 5e+37:
		tmp = t_1 * -y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	t_2 = Float64(0.5 * Float64(x * t_1))
	tmp = 0.0
	if (Float64(x * 0.5) <= -5e+18)
		tmp = t_2;
	elseif (Float64(x * 0.5) <= 5e+37)
		tmp = Float64(t_1 * Float64(-y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	t_2 = 0.5 * (x * t_1);
	tmp = 0.0;
	if ((x * 0.5) <= -5e+18)
		tmp = t_2;
	elseif ((x * 0.5) <= 5e+37)
		tmp = t_1 * -y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * 0.5), $MachinePrecision], -5e+18], t$95$2, If[LessEqual[N[(x * 0.5), $MachinePrecision], 5e+37], N[(t$95$1 * (-y)), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -5e18 or 4.99999999999999989e37 < (*.f64 x #s(literal 1/2 binary64))

   1. Initial program 99.8%

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

   3. 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 57.8%

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

   6. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. unsub-neg N/A

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

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

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

      15. /-lowering-/.f64 57.8

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

   8. Simplified 57.8%

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

      1. *-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 57.8

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

   10. Applied egg-rr 57.8%

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

   11. Taylor expanded in y around 0

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

   13. Simplified 46.6%

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

   if -5e18 < (*.f64 x #s(literal 1/2 binary64)) < 4.99999999999999989e37

   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 54.1%

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

   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 47.3

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

   8. Simplified 47.3%

      \[\leadsto \left(\color{blue}{\left(-y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]
   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. *-lowering-*.f64 N/A

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

      6. neg-lowering-neg.f64 47.3

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

   10. Applied egg-rr 47.3%

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

Alternative 13: 85.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 85.5

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

5. Simplified 85.5%

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

Alternative 14: 57.5% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $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 55.7%

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

   1. *-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. *-lowering-*.f64 55.7

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

7. Applied egg-rr 55.7%

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

Alternative 15: 29.7% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * (-y)), $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 55.7%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]
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 33.6

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

8. Simplified 33.6%

   \[\leadsto \left(\color{blue}{\left(-y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]
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. *-lowering-*.f64 N/A

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

   6. neg-lowering-neg.f64 33.6

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

10. Applied egg-rr 33.6%

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

Developer Target 1: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024204 
(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))))