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

Percentage Accurate: 99.5% → 99.5%

Time: 16.9s

Alternatives: 10

Speedup: 1.0×

• 44.2% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-/.f64 N/A

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

   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. lower-*.f64 99.4

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

4. Applied rewrites 99.4%

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

5. Final simplification 99.4%

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

Alternative 2: 95.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \]
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 95.3

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

5. Applied rewrites 95.3%

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

6. Final simplification 95.3%

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

Alternative 3: 95.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \]
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 95.3

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

5. Applied rewrites 95.3%

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

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 95.3%

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

9. Final simplification 95.3%

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

Alternative 4: 94.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

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

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 94.2

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 94.2

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 94.2

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

7. Applied rewrites 94.2%

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

8. Final simplification 94.2%

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

Alternative 5: 88.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double t_2 = sqrt((2.0 * z));
	double tmp;
	if ((t * t) <= 2.0) {
		tmp = t_1 * t_2;
	} else {
		tmp = (((t * t) * 0.5) * t_1) * 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 = (0.5d0 * x) - y
    t_2 = sqrt((2.0d0 * z))
    if ((t * t) <= 2.0d0) then
        tmp = t_1 * t_2
    else
        tmp = (((t * t) * 0.5d0) * t_1) * t_2
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

   3. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
   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 99.5

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

   5. Applied rewrites 99.5%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, 0.5, 1\right)} \]
   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 99.5

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

      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 99.5

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

      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 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.2

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

   10. Applied rewrites 99.2%

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

   if 2 < (*.f64 t t)

   1. Initial program 99.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-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 73.6

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

   5. Applied rewrites 73.6%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, 0.5, 1\right)} \]
   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 77.3

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

      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 77.3

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

      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 77.3

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

   7. Applied rewrites 77.3%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 77.3%

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

4. Final simplification 88.0%

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

Alternative 6: 75.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \cdot t \leq 1.32 \cdot 10^{-11}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (<= (* t t) 1.32e-11)
     (* (- (* 0.5 x) y) t_1)
     (* (* (- y) (fma (* t t) 0.5 1.0)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if ((t * t) <= 1.32e-11) {
		tmp = ((0.5 * x) - y) * t_1;
	} else {
		tmp = (-y * fma((t * t), 0.5, 1.0)) * t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (Float64(t * t) <= 1.32e-11)
		tmp = Float64(Float64(Float64(0.5 * x) - y) * t_1);
	else
		tmp = Float64(Float64(Float64(-y) * fma(Float64(t * t), 0.5, 1.0)) * t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 1.32e-11], N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[((-y) * N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 1.32e-11

   1. Initial program 99.5%

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

   3. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
   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 99.5

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

   5. Applied rewrites 99.5%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, 0.5, 1\right)} \]
   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 99.5

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

      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 99.5

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

      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 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.4

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

   10. Applied rewrites 99.4%

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

   if 1.32e-11 < (*.f64 t t)

   1. Initial program 99.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-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 73.8

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

   5. Applied rewrites 73.8%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, 0.5, 1\right)} \]
   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 77.4

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

      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 77.4

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

      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 77.4

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

   7. Applied rewrites 77.4%

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

   8. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 57.4

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

   10. Applied rewrites 57.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 1.32 \cdot 10^{-11}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\right) \cdot \sqrt{2 \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.3% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
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 86.4

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

5. Applied rewrites 86.4%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, 0.5, 1\right)} \]
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 88.2

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

   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 88.2

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

   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 88.2

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

7. Applied rewrites 88.2%

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

8. Taylor expanded in t around 0

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

   1. sub-neg N/A

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

   2. distribute-lft-out N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-neg.f64 88.2

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

10. Applied rewrites 88.2%

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

11. Final simplification 88.2%

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

Alternative 8: 88.3% 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 99.4%

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

3. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
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 86.4

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

5. Applied rewrites 86.4%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, 0.5, 1\right)} \]
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 88.2

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

   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 88.2

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

   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 88.2

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

7. Applied rewrites 88.2%

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

Alternative 9: 57.6% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
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 86.4

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

5. Applied rewrites 86.4%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, 0.5, 1\right)} \]
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 88.2

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

   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 88.2

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

   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 88.2

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

7. Applied rewrites 88.2%

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

8. Taylor expanded in t around 0

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

   1. lower--.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 56.7

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

10. Applied rewrites 56.7%

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

11. Final simplification 56.7%

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

Alternative 10: 30.1% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in t around 0

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

5. Applied rewrites 56.7%

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

6. Taylor expanded in y around inf

   \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]
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 30.9

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

8. Applied rewrites 30.9%

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

9. Final simplification 30.9%

   \[\leadsto 1 \cdot \left(\left(-y\right) \cdot \sqrt{2 \cdot z}\right) \]
10. 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 2024243 
(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))))