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

Percentage Accurate: 99.4% → 99.4%

Time: 10.6s

Alternatives: 10

Speedup: 1.0×

• 44.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) * 0.5d0))
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) * 0.5));
}

Python

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

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) * 0.5)))
end

MATLAB

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

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

   2. lift-/.f64 N/A

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

   3. exp-sqrt N/A

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

   4. pow1/2 N/A

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

   5. pow-exp N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-*.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 rewrites 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. Add Preprocessing

Alternative 2: 94.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 95.0

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

5. Applied rewrites 95.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 95.3

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

7. Applied rewrites 95.3%

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

9. Applied rewrites 95.3%

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

Alternative 3: 94.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 95.0

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

5. Applied rewrites 95.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 95.3

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

7. Applied rewrites 95.3%

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

8. Taylor expanded in t around inf

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

10. Applied rewrites 95.3%

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

11. Final simplification 95.3%

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

Alternative 4: 94.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * fma(fma(Float64(0.020833333333333332 * Float64(t * t)), Float64(t * t), 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[(N[(N[(0.020833333333333332 * N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * 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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 95.0

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

5. Applied rewrites 95.0%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 95.0%

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

Alternative 5: 93.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 92.4

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

5. Applied rewrites 92.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 92.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 92.8

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

7. Applied rewrites 92.8%

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

Alternative 6: 92.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 92.4

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

5. Applied rewrites 92.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 92.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 92.8

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

7. Applied rewrites 92.8%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 92.8

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

9. Applied rewrites 92.8%

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

Alternative 7: 64.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 2.8 \cdot 10^{+74}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot t\_1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot t\_1\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= t 2.8e+74)
     (* (* (- (* x 0.5) y) t_1) 1.0)
     (* (* (- y) t_1) (fma (* t t) 0.5 1.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (t <= 2.8e+74) {
		tmp = (((x * 0.5) - y) * t_1) * 1.0;
	} else {
		tmp = (-y * t_1) * fma((t * t), 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 2.8e+74)
		tmp = Float64(Float64(Float64(Float64(x * 0.5) - y) * t_1) * 1.0);
	else
		tmp = Float64(Float64(Float64(-y) * t_1) * fma(Float64(t * t), 0.5, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 2.8e+74], N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[((-y) * t$95$1), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.80000000000000002e74

   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. Applied rewrites 68.1%

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

   if 2.80000000000000002e74 < 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 89.5

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.9

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

   8. Applied rewrites 69.9%

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

Alternative 8: 87.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 85.7

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

5. Applied rewrites 85.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 87.2

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 87.2

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 87.2

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

7. Applied rewrites 87.2%

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

Alternative 9: 56.0% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

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))) * 1.0d0
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))) * 1.0;
}

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * 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] * 1.0), $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. Applied rewrites 59.7%

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

Alternative 10: 29.2% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

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 = (-y * sqrt((z * 2.0d0))) * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[((-y) * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $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. Applied rewrites 59.7%

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

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. lower-neg.f64 29.9

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

8. Applied rewrites 29.9%

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

Developer Target 1: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024339 
(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))))