Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, C

Percentage Accurate: 99.9% → 99.9%

Time: 9.1s

Alternatives: 11

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(x + \sin y\right) + z \cdot \cos y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ x (sin y)) (* z (cos y))))

C

double code(double x, double y, double z) {
	return (x + sin(y)) + (z * cos(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + sin(y)) + (z * cos(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x + Math.sin(y)) + (z * Math.cos(y));
}

Python

def code(x, y, z):
	return (x + math.sin(y)) + (z * math.cos(y))

Julia

function code(x, y, z)
	return Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + sin(y)) + (z * cos(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + \sin y\right) + z \cdot \cos y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ x (sin y)) (* z (cos y))))

C

double code(double x, double y, double z) {
	return (x + sin(y)) + (z * cos(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + sin(y)) + (z * cos(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x + Math.sin(y)) + (z * Math.cos(y));
}

Python

def code(x, y, z):
	return (x + math.sin(y)) + (z * math.cos(y))

Julia

function code(x, y, z)
	return Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + sin(y)) + (z * cos(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin y + \mathsf{fma}\left(\cos y, z, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (sin y) (fma (cos y) z x)))

C

double code(double x, double y, double z) {
	return sin(y) + fma(cos(y), z, x);
}

Julia

function code(x, y, z)
	return Float64(sin(y) + fma(cos(y), z, x))
end

Wolfram

code[x_, y_, z_] := N[(N[Sin[y], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * z + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(x + \sin y\right) + z \cdot \cos y \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(x + \sin y\right) + z \cdot \cos y} \]

   2. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(x + \sin y\right)} + z \cdot \cos y \]

   3. associate-+l+ N/A

      \[\leadsto \color{blue}{x + \left(\sin y + z \cdot \cos y\right)} \]

   4. +-commutative N/A

      \[\leadsto x + \color{blue}{\left(z \cdot \cos y + \sin y\right)} \]

   5. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(x + z \cdot \cos y\right) + \sin y} \]

   6. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(x + z \cdot \cos y\right) + \sin y} \]

   7. +-commutative N/A

      \[\leadsto \color{blue}{\left(z \cdot \cos y + x\right)} + \sin y \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{z \cdot \cos y} + x\right) + \sin y \]

   9. *-commutative N/A

      \[\leadsto \left(\color{blue}{\cos y \cdot z} + x\right) + \sin y \]

   10. lower-fma.f64 99.9

       \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, x\right)} + \sin y \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, x\right) + \sin y} \]

5. Final simplification 99.9%

   \[\leadsto \sin y + \mathsf{fma}\left(\cos y, z, x\right) \]
6. Add Preprocessing

Alternative 2: 80.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y + \left(\sin y + x\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+16}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\left(x + y\right) + z\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* z (cos y)) (+ (sin y) x))))
   (if (<= t_0 -5e+16)
     (+ x z)
     (if (<= t_0 -0.02)
       (sin y)
       (if (<= t_0 5e-20) (+ (+ x y) z) (if (<= t_0 1.0) (sin y) (+ x z)))))))

C

double code(double x, double y, double z) {
	double t_0 = (z * cos(y)) + (sin(y) + x);
	double tmp;
	if (t_0 <= -5e+16) {
		tmp = x + z;
	} else if (t_0 <= -0.02) {
		tmp = sin(y);
	} else if (t_0 <= 5e-20) {
		tmp = (x + y) + z;
	} else if (t_0 <= 1.0) {
		tmp = sin(y);
	} else {
		tmp = x + z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z * cos(y)) + (sin(y) + x)
    if (t_0 <= (-5d+16)) then
        tmp = x + z
    else if (t_0 <= (-0.02d0)) then
        tmp = sin(y)
    else if (t_0 <= 5d-20) then
        tmp = (x + y) + z
    else if (t_0 <= 1.0d0) then
        tmp = sin(y)
    else
        tmp = x + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z * Math.cos(y)) + (Math.sin(y) + x);
	double tmp;
	if (t_0 <= -5e+16) {
		tmp = x + z;
	} else if (t_0 <= -0.02) {
		tmp = Math.sin(y);
	} else if (t_0 <= 5e-20) {
		tmp = (x + y) + z;
	} else if (t_0 <= 1.0) {
		tmp = Math.sin(y);
	} else {
		tmp = x + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * math.cos(y)) + (math.sin(y) + x)
	tmp = 0
	if t_0 <= -5e+16:
		tmp = x + z
	elif t_0 <= -0.02:
		tmp = math.sin(y)
	elif t_0 <= 5e-20:
		tmp = (x + y) + z
	elif t_0 <= 1.0:
		tmp = math.sin(y)
	else:
		tmp = x + z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * cos(y)) + Float64(sin(y) + x))
	tmp = 0.0
	if (t_0 <= -5e+16)
		tmp = Float64(x + z);
	elseif (t_0 <= -0.02)
		tmp = sin(y);
	elseif (t_0 <= 5e-20)
		tmp = Float64(Float64(x + y) + z);
	elseif (t_0 <= 1.0)
		tmp = sin(y);
	else
		tmp = Float64(x + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * cos(y)) + (sin(y) + x);
	tmp = 0.0;
	if (t_0 <= -5e+16)
		tmp = x + z;
	elseif (t_0 <= -0.02)
		tmp = sin(y);
	elseif (t_0 <= 5e-20)
		tmp = (x + y) + z;
	elseif (t_0 <= 1.0)
		tmp = sin(y);
	else
		tmp = x + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+16], N[(x + z), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Sin[y], $MachinePrecision], If[LessEqual[t$95$0, 5e-20], N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[y], $MachinePrecision], N[(x + z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -5e16 or 1 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + z} \]
   4. Step-by-step derivation

      1. lower-+.f64 81.5

         \[\leadsto \color{blue}{x + z} \]

   5. Applied rewrites 81.5%

      \[\leadsto \color{blue}{x + z} \]

   if -5e16 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.0200000000000000004 or 4.9999999999999999e-20 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 1

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \sin y\right) + z \cdot \cos y} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \sin y\right)} + z \cdot \cos y \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x + \left(\sin y + z \cdot \cos y\right)} \]

      4. +-commutative N/A

         \[\leadsto x + \color{blue}{\left(z \cdot \cos y + \sin y\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + z \cdot \cos y\right) + \sin y} \]

      6. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + z \cdot \cos y\right) + \sin y} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \cos y + x\right)} + \sin y \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{z \cdot \cos y} + x\right) + \sin y \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\cos y \cdot z} + x\right) + \sin y \]

      10. lower-fma.f64 99.9

          \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, x\right)} + \sin y \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, x\right) + \sin y} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\sin y + z \cdot \cos y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \cos y + \sin y} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\cos y \cdot z} + \sin y \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, \sin y\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\cos y}, z, \sin y\right) \]

      5. lower-sin.f64 93.6

         \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{\sin y}\right) \]

   7. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, \sin y\right)} \]

   8. Taylor expanded in z around 0

      \[\leadsto \sin y \]
   9. Step-by-step derivation

   10. Applied rewrites 91.4%

       \[\leadsto \sin y \]

   if -0.0200000000000000004 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 4.9999999999999999e-20

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \left(y + z\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + y\right) + z} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + y\right) + z} \]

      3. lower-+.f64 100.0

         \[\leadsto \color{blue}{\left(x + y\right)} + z \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(x + y\right) + z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \cos y + \left(\sin y + x\right) \leq -5 \cdot 10^{+16}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \cdot \cos y + \left(\sin y + x\right) \leq -0.02:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;z \cdot \cos y + \left(\sin y + x\right) \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\left(x + y\right) + z\\ \mathbf{elif}\;z \cdot \cos y + \left(\sin y + x\right) \leq 1:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y + \left(\sin y + x\right)\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;x + z\\ \mathbf{elif}\;t\_0 \leq 0.15:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot y + \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* z (cos y)) (+ (sin y) x))))
   (if (<= t_0 -0.1)
     (+ x z)
     (if (<= t_0 0.15)
       (+
        (*
         (fma
          (fma 0.008333333333333333 (* y y) -0.16666666666666666)
          (* y y)
          1.0)
         y)
        (+ x z))
       (+ x z)))))

C

double code(double x, double y, double z) {
	double t_0 = (z * cos(y)) + (sin(y) + x);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = x + z;
	} else if (t_0 <= 0.15) {
		tmp = (fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0) * y) + (x + z);
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * cos(y)) + Float64(sin(y) + x))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(x + z);
	elseif (t_0 <= 0.15)
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * y) + Float64(x + z));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], N[(x + z), $MachinePrecision], If[LessEqual[t$95$0, 0.15], N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] + N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.10000000000000001 or 0.149999999999999994 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + z} \]
   4. Step-by-step derivation

      1. lower-+.f64 68.8

         \[\leadsto \color{blue}{x + z} \]

   5. Applied rewrites 68.8%

      \[\leadsto \color{blue}{x + z} \]

   if -0.10000000000000001 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 0.149999999999999994

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \sin y\right) + z \cdot \cos y} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \sin y\right)} + z \cdot \cos y \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x + \left(\sin y + z \cdot \cos y\right)} \]

      4. +-commutative N/A

         \[\leadsto x + \color{blue}{\left(z \cdot \cos y + \sin y\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + z \cdot \cos y\right) + \sin y} \]

      6. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + z \cdot \cos y\right) + \sin y} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \cos y + x\right)} + \sin y \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{z \cdot \cos y} + x\right) + \sin y \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\cos y \cdot z} + x\right) + \sin y \]

      10. lower-fma.f64 100.0

          \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, x\right)} + \sin y \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, x\right) + \sin y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + z\right)} + \sin y \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + x\right)} + \sin y \]

      2. lower-+.f64 100.0

         \[\leadsto \color{blue}{\left(z + x\right)} + \sin y \]

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(z + x\right)} + \sin y \]

   8. Taylor expanded in y around 0

      \[\leadsto \left(z + x\right) + \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(z + x\right) + \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(z + x\right) + \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot y} \]

      3. +-commutative N/A

         \[\leadsto \left(z + x\right) + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) + 1\right)} \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(z + x\right) + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) \cdot {y}^{2}} + 1\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(z + x\right) + \color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right)} \cdot y \]

      6. sub-neg N/A

         \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {y}^{2}, 1\right) \cdot y \]

      7. metadata-eval N/A

         \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} + \color{blue}{\frac{-1}{6}}, {y}^{2}, 1\right) \cdot y \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {y}^{2}, \frac{-1}{6}\right)}, {y}^{2}, 1\right) \cdot y \]

      9. unpow2 N/A

         \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{y \cdot y}, \frac{-1}{6}\right), {y}^{2}, 1\right) \cdot y \]

      10. lower-*.f64 N/A

          \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{y \cdot y}, \frac{-1}{6}\right), {y}^{2}, 1\right) \cdot y \]

      11. unpow2 N/A

          \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \color{blue}{y \cdot y}, 1\right) \cdot y \]

      12. lower-*.f64 93.5

          \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right) \cdot y \]

   10. Applied rewrites 93.5%

       \[\leadsto \left(z + x\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \cos y + \left(\sin y + x\right) \leq -0.1:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \cdot \cos y + \left(\sin y + x\right) \leq 0.15:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot y + \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y + \left(\sin y + x\right)\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;x + z\\ \mathbf{elif}\;t\_0 \leq 0.15:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* z (cos y)) (+ (sin y) x))))
   (if (<= t_0 -0.5)
     (+ x z)
     (if (<= t_0 0.15) (fma (fma (* -0.5 y) z 1.0) y (+ x z)) (+ x z)))))

C

double code(double x, double y, double z) {
	double t_0 = (z * cos(y)) + (sin(y) + x);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = x + z;
	} else if (t_0 <= 0.15) {
		tmp = fma(fma((-0.5 * y), z, 1.0), y, (x + z));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * cos(y)) + Float64(sin(y) + x))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(x + z);
	elseif (t_0 <= 0.15)
		tmp = fma(fma(Float64(-0.5 * y), z, 1.0), y, Float64(x + z));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(x + z), $MachinePrecision], If[LessEqual[t$95$0, 0.15], N[(N[(N[(-0.5 * y), $MachinePrecision] * z + 1.0), $MachinePrecision] * y + N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.5 or 0.149999999999999994 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + z} \]
   4. Step-by-step derivation

      1. lower-+.f64 70.6

         \[\leadsto \color{blue}{x + z} \]

   5. Applied rewrites 70.6%

      \[\leadsto \color{blue}{x + z} \]

   if -0.5 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 0.149999999999999994

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 78.1

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

   5. Applied rewrites 78.1%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \cos y + \left(\sin y + x\right) \leq -0.5:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \cdot \cos y + \left(\sin y + x\right) \leq 0.15:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y + \left(\sin y + x\right)\\ \mathbf{if}\;t\_0 \leq -0.16:\\ \;\;\;\;x + z\\ \mathbf{elif}\;t\_0 \leq 0.15:\\ \;\;\;\;\left(x + y\right) + z\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* z (cos y)) (+ (sin y) x))))
   (if (<= t_0 -0.16) (+ x z) (if (<= t_0 0.15) (+ (+ x y) z) (+ x z)))))

C

double code(double x, double y, double z) {
	double t_0 = (z * cos(y)) + (sin(y) + x);
	double tmp;
	if (t_0 <= -0.16) {
		tmp = x + z;
	} else if (t_0 <= 0.15) {
		tmp = (x + y) + z;
	} else {
		tmp = x + z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z * cos(y)) + (sin(y) + x)
    if (t_0 <= (-0.16d0)) then
        tmp = x + z
    else if (t_0 <= 0.15d0) then
        tmp = (x + y) + z
    else
        tmp = x + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z * Math.cos(y)) + (Math.sin(y) + x);
	double tmp;
	if (t_0 <= -0.16) {
		tmp = x + z;
	} else if (t_0 <= 0.15) {
		tmp = (x + y) + z;
	} else {
		tmp = x + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * math.cos(y)) + (math.sin(y) + x)
	tmp = 0
	if t_0 <= -0.16:
		tmp = x + z
	elif t_0 <= 0.15:
		tmp = (x + y) + z
	else:
		tmp = x + z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * cos(y)) + Float64(sin(y) + x))
	tmp = 0.0
	if (t_0 <= -0.16)
		tmp = Float64(x + z);
	elseif (t_0 <= 0.15)
		tmp = Float64(Float64(x + y) + z);
	else
		tmp = Float64(x + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * cos(y)) + (sin(y) + x);
	tmp = 0.0;
	if (t_0 <= -0.16)
		tmp = x + z;
	elseif (t_0 <= 0.15)
		tmp = (x + y) + z;
	else
		tmp = x + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.16], N[(x + z), $MachinePrecision], If[LessEqual[t$95$0, 0.15], N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision], N[(x + z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.160000000000000003 or 0.149999999999999994 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + z} \]
   4. Step-by-step derivation

      1. lower-+.f64 69.1

         \[\leadsto \color{blue}{x + z} \]

   5. Applied rewrites 69.1%

      \[\leadsto \color{blue}{x + z} \]

   if -0.160000000000000003 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 0.149999999999999994

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \left(y + z\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + y\right) + z} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + y\right) + z} \]

      3. lower-+.f64 89.5

         \[\leadsto \color{blue}{\left(x + y\right)} + z \]

   5. Applied rewrites 89.5%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \cos y + \left(\sin y + x\right) \leq -0.16:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \cdot \cos y + \left(\sin y + x\right) \leq 0.15:\\ \;\;\;\;\left(x + y\right) + z\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -1 \cdot 10^{+207}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.32 \cdot 10^{-33}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-28}:\\ \;\;\;\;\sin y + x\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+151}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -1e+207)
     t_0
     (if (<= z -1.32e-33)
       (+ x z)
       (if (<= z 1.35e-28) (+ (sin y) x) (if (<= z 8.2e+151) (+ x z) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -1e+207) {
		tmp = t_0;
	} else if (z <= -1.32e-33) {
		tmp = x + z;
	} else if (z <= 1.35e-28) {
		tmp = sin(y) + x;
	} else if (z <= 8.2e+151) {
		tmp = x + z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * cos(y)
    if (z <= (-1d+207)) then
        tmp = t_0
    else if (z <= (-1.32d-33)) then
        tmp = x + z
    else if (z <= 1.35d-28) then
        tmp = sin(y) + x
    else if (z <= 8.2d+151) then
        tmp = x + z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.cos(y);
	double tmp;
	if (z <= -1e+207) {
		tmp = t_0;
	} else if (z <= -1.32e-33) {
		tmp = x + z;
	} else if (z <= 1.35e-28) {
		tmp = Math.sin(y) + x;
	} else if (z <= 8.2e+151) {
		tmp = x + z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	tmp = 0
	if z <= -1e+207:
		tmp = t_0
	elif z <= -1.32e-33:
		tmp = x + z
	elif z <= 1.35e-28:
		tmp = math.sin(y) + x
	elif z <= 8.2e+151:
		tmp = x + z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -1e+207)
		tmp = t_0;
	elseif (z <= -1.32e-33)
		tmp = Float64(x + z);
	elseif (z <= 1.35e-28)
		tmp = Float64(sin(y) + x);
	elseif (z <= 8.2e+151)
		tmp = Float64(x + z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	tmp = 0.0;
	if (z <= -1e+207)
		tmp = t_0;
	elseif (z <= -1.32e-33)
		tmp = x + z;
	elseif (z <= 1.35e-28)
		tmp = sin(y) + x;
	elseif (z <= 8.2e+151)
		tmp = x + z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+207], t$95$0, If[LessEqual[z, -1.32e-33], N[(x + z), $MachinePrecision], If[LessEqual[z, 1.35e-28], N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 8.2e+151], N[(x + z), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1e207 or 8.1999999999999996e151 < z

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \cos y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\cos y \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\cos y \cdot z} \]

      3. lower-cos.f64 94.2

         \[\leadsto \color{blue}{\cos y} \cdot z \]

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\cos y \cdot z} \]

   if -1e207 < z < -1.31999999999999993e-33 or 1.3499999999999999e-28 < z < 8.1999999999999996e151

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + z} \]
   4. Step-by-step derivation

      1. lower-+.f64 83.4

         \[\leadsto \color{blue}{x + z} \]

   5. Applied rewrites 83.4%

      \[\leadsto \color{blue}{x + z} \]

   if -1.31999999999999993e-33 < z < 1.3499999999999999e-28

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \sin y} \]
   4. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \sin y} \]

      2. lower-sin.f64 95.2

         \[\leadsto x + \color{blue}{\sin y} \]

   5. Applied rewrites 95.2%

      \[\leadsto \color{blue}{x + \sin y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+207}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq -1.32 \cdot 10^{-33}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-28}:\\ \;\;\;\;\sin y + x\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+151}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -1 \cdot 10^{+207}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+151}:\\ \;\;\;\;\left(x + z\right) + \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -1e+207) t_0 (if (<= z 8.2e+151) (+ (+ x z) (sin y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -1e+207) {
		tmp = t_0;
	} else if (z <= 8.2e+151) {
		tmp = (x + z) + sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * cos(y)
    if (z <= (-1d+207)) then
        tmp = t_0
    else if (z <= 8.2d+151) then
        tmp = (x + z) + sin(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.cos(y);
	double tmp;
	if (z <= -1e+207) {
		tmp = t_0;
	} else if (z <= 8.2e+151) {
		tmp = (x + z) + Math.sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	tmp = 0
	if z <= -1e+207:
		tmp = t_0
	elif z <= 8.2e+151:
		tmp = (x + z) + math.sin(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -1e+207)
		tmp = t_0;
	elseif (z <= 8.2e+151)
		tmp = Float64(Float64(x + z) + sin(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	tmp = 0.0;
	if (z <= -1e+207)
		tmp = t_0;
	elseif (z <= 8.2e+151)
		tmp = (x + z) + sin(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+207], t$95$0, If[LessEqual[z, 8.2e+151], N[(N[(x + z), $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1e207 or 8.1999999999999996e151 < z

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \cos y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\cos y \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\cos y \cdot z} \]

      3. lower-cos.f64 94.2

         \[\leadsto \color{blue}{\cos y} \cdot z \]

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\cos y \cdot z} \]

   if -1e207 < z < 8.1999999999999996e151

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \sin y\right) + z \cdot \cos y} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \sin y\right)} + z \cdot \cos y \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x + \left(\sin y + z \cdot \cos y\right)} \]

      4. +-commutative N/A

         \[\leadsto x + \color{blue}{\left(z \cdot \cos y + \sin y\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + z \cdot \cos y\right) + \sin y} \]

      6. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + z \cdot \cos y\right) + \sin y} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \cos y + x\right)} + \sin y \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{z \cdot \cos y} + x\right) + \sin y \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\cos y \cdot z} + x\right) + \sin y \]

      10. lower-fma.f64 99.9

          \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, x\right)} + \sin y \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, x\right) + \sin y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + z\right)} + \sin y \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + x\right)} + \sin y \]

      2. lower-+.f64 93.4

         \[\leadsto \color{blue}{\left(z + x\right)} + \sin y \]

   7. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\left(z + x\right)} + \sin y \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+207}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+151}:\\ \;\;\;\;\left(x + z\right) + \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y + x\\ \mathbf{if}\;y \leq -6 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 55000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot y + \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (sin y) x)))
   (if (<= y -6e+26)
     t_0
     (if (<= y 55000000.0)
       (+
        (*
         (fma
          (fma 0.008333333333333333 (* y y) -0.16666666666666666)
          (* y y)
          1.0)
         y)
        (+ x z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) + x;
	double tmp;
	if (y <= -6e+26) {
		tmp = t_0;
	} else if (y <= 55000000.0) {
		tmp = (fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0) * y) + (x + z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) + x)
	tmp = 0.0
	if (y <= -6e+26)
		tmp = t_0;
	elseif (y <= 55000000.0)
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * y) + Float64(x + z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -6e+26], t$95$0, If[LessEqual[y, 55000000.0], N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] + N[(x + z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.99999999999999994e26 or 5.5e7 < y

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \sin y} \]
   4. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \color{blue}{x + \sin y} \]

      2. lower-sin.f64 69.6

         \[\leadsto x + \color{blue}{\sin y} \]

   5. Applied rewrites 69.6%

      \[\leadsto \color{blue}{x + \sin y} \]

   if -5.99999999999999994e26 < y < 5.5e7

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \sin y\right) + z \cdot \cos y} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \sin y\right)} + z \cdot \cos y \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x + \left(\sin y + z \cdot \cos y\right)} \]

      4. +-commutative N/A

         \[\leadsto x + \color{blue}{\left(z \cdot \cos y + \sin y\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + z \cdot \cos y\right) + \sin y} \]

      6. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + z \cdot \cos y\right) + \sin y} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \cos y + x\right)} + \sin y \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{z \cdot \cos y} + x\right) + \sin y \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\cos y \cdot z} + x\right) + \sin y \]

      10. lower-fma.f64 100.0

          \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, x\right)} + \sin y \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, x\right) + \sin y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + z\right)} + \sin y \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + x\right)} + \sin y \]

      2. lower-+.f64 98.2

         \[\leadsto \color{blue}{\left(z + x\right)} + \sin y \]

   7. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\left(z + x\right)} + \sin y \]

   8. Taylor expanded in y around 0

      \[\leadsto \left(z + x\right) + \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(z + x\right) + \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(z + x\right) + \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot y} \]

      3. +-commutative N/A

         \[\leadsto \left(z + x\right) + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) + 1\right)} \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(z + x\right) + \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) \cdot {y}^{2}} + 1\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(z + x\right) + \color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right)} \cdot y \]

      6. sub-neg N/A

         \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {y}^{2}, 1\right) \cdot y \]

      7. metadata-eval N/A

         \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} + \color{blue}{\frac{-1}{6}}, {y}^{2}, 1\right) \cdot y \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {y}^{2}, \frac{-1}{6}\right)}, {y}^{2}, 1\right) \cdot y \]

      9. unpow2 N/A

         \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{y \cdot y}, \frac{-1}{6}\right), {y}^{2}, 1\right) \cdot y \]

      10. lower-*.f64 N/A

          \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{y \cdot y}, \frac{-1}{6}\right), {y}^{2}, 1\right) \cdot y \]

      11. unpow2 N/A

          \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \color{blue}{y \cdot y}, 1\right) \cdot y \]

      12. lower-*.f64 98.1

          \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right) \cdot y \]

   10. Applied rewrites 98.1%

       \[\leadsto \left(z + x\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+26}:\\ \;\;\;\;\sin y + x\\ \mathbf{elif}\;y \leq 55000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot y + \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4200000000:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot y + \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4200000000.0)
   (+ x z)
   (if (<= y 4.8e+18)
     (+
      (*
       (fma
        (fma
         (fma -0.0001984126984126984 (* y y) 0.008333333333333333)
         (* y y)
         -0.16666666666666666)
        (* y y)
        1.0)
       y)
      (+ x z))
     (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4200000000.0) {
		tmp = x + z;
	} else if (y <= 4.8e+18) {
		tmp = (fma(fma(fma(-0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), -0.16666666666666666), (y * y), 1.0) * y) + (x + z);
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4200000000.0)
		tmp = Float64(x + z);
	elseif (y <= 4.8e+18)
		tmp = Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * y) + Float64(x + z));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4200000000.0], N[(x + z), $MachinePrecision], If[LessEqual[y, 4.8e+18], N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] + N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2e9 or 4.8e18 < y

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + z} \]
   4. Step-by-step derivation

      1. lower-+.f64 43.1

         \[\leadsto \color{blue}{x + z} \]

   5. Applied rewrites 43.1%

      \[\leadsto \color{blue}{x + z} \]

   if -4.2e9 < y < 4.8e18

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \sin y\right) + z \cdot \cos y} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \sin y\right)} + z \cdot \cos y \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x + \left(\sin y + z \cdot \cos y\right)} \]

      4. +-commutative N/A

         \[\leadsto x + \color{blue}{\left(z \cdot \cos y + \sin y\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + z \cdot \cos y\right) + \sin y} \]

      6. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + z \cdot \cos y\right) + \sin y} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \cos y + x\right)} + \sin y \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{z \cdot \cos y} + x\right) + \sin y \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{\cos y \cdot z} + x\right) + \sin y \]

      10. lower-fma.f64 100.0

          \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, x\right)} + \sin y \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, x\right) + \sin y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + z\right)} + \sin y \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + x\right)} + \sin y \]

      2. lower-+.f64 99.4

         \[\leadsto \color{blue}{\left(z + x\right)} + \sin y \]

   7. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(z + x\right)} + \sin y \]

   8. Taylor expanded in y around 0

      \[\leadsto \left(z + x\right) + \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(z + x\right) + \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(z + x\right) + \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right) \cdot y} \]

      3. +-commutative N/A

         \[\leadsto \left(z + x\right) + \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) + 1\right)} \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left(z + x\right) + \left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot {y}^{2}} + 1\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(z + x\right) + \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}, {y}^{2}, 1\right)} \cdot y \]

      6. sub-neg N/A

         \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {y}^{2}, 1\right) \cdot y \]

      7. *-commutative N/A

         \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {y}^{2}, 1\right) \cdot y \]

      8. metadata-eval N/A

         \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} + \color{blue}{\frac{-1}{6}}, {y}^{2}, 1\right) \cdot y \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}, {y}^{2}, \frac{-1}{6}\right)}, {y}^{2}, 1\right) \cdot y \]

      10. +-commutative N/A

          \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{5040} \cdot {y}^{2} + \frac{1}{120}}, {y}^{2}, \frac{-1}{6}\right), {y}^{2}, 1\right) \cdot y \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{5040}, {y}^{2}, \frac{1}{120}\right)}, {y}^{2}, \frac{-1}{6}\right), {y}^{2}, 1\right) \cdot y \]

      12. unpow2 N/A

          \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \color{blue}{y \cdot y}, \frac{1}{120}\right), {y}^{2}, \frac{-1}{6}\right), {y}^{2}, 1\right) \cdot y \]

      13. lower-*.f64 N/A

          \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \color{blue}{y \cdot y}, \frac{1}{120}\right), {y}^{2}, \frac{-1}{6}\right), {y}^{2}, 1\right) \cdot y \]

      14. unpow2 N/A

          \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), \color{blue}{y \cdot y}, \frac{-1}{6}\right), {y}^{2}, 1\right) \cdot y \]

      15. lower-*.f64 N/A

          \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), \color{blue}{y \cdot y}, \frac{-1}{6}\right), {y}^{2}, 1\right) \cdot y \]

      16. unpow2 N/A

          \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{-1}{6}\right), \color{blue}{y \cdot y}, 1\right) \cdot y \]

      17. lower-*.f64 98.7

          \[\leadsto \left(z + x\right) + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right) \cdot y \]

   10. Applied rewrites 98.7%

       \[\leadsto \left(z + x\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4200000000:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot y + \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -460000:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -460000.0)
   (+ x z)
   (if (<= y 7.5e-25)
     (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) y (+ x z))
     (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -460000.0) {
		tmp = x + z;
	} else if (y <= 7.5e-25) {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, (x + z));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -460000.0)
		tmp = Float64(x + z);
	elseif (y <= 7.5e-25)
		tmp = fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, Float64(x + z));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -460000.0], N[(x + z), $MachinePrecision], If[LessEqual[y, 7.5e-25], N[(N[(N[(-0.16666666666666666 * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.6e5 or 7.49999999999999989e-25 < y

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + z} \]
   4. Step-by-step derivation

      1. lower-+.f64 44.8

         \[\leadsto \color{blue}{x + z} \]

   5. Applied rewrites 44.8%

      \[\leadsto \color{blue}{x + z} \]

   if -4.6e5 < y < 7.49999999999999989e-25

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + z\right) + y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right) + \left(x + z\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right) \cdot y} + \left(x + z\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right), y, x + z\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right) + 1}, y, x + z\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right) \cdot y} + 1, y, x + z\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y, y, 1\right)}, y, x + z\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{6} \cdot y + \frac{-1}{2} \cdot z}, y, 1\right), y, x + z\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, y, \frac{-1}{2} \cdot z\right)}, y, 1\right), y, x + z\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, y, \color{blue}{\frac{-1}{2} \cdot z}\right), y, 1\right), y, x + z\right) \]

      11. lower-+.f64 99.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, \color{blue}{x + z}\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, x + z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 66.8% accurate, 53.0× speedup

Math

\[\begin{array}{l} \\ x + z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x z))

C

double code(double x, double y, double z) {
	return x + z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + z
end function

Java

public static double code(double x, double y, double z) {
	return x + z;
}

Python

def code(x, y, z):
	return x + z

Julia

function code(x, y, z)
	return Float64(x + z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + z;
end

Wolfram

code[x_, y_, z_] := N[(x + z), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(x + \sin y\right) + z \cdot \cos y \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + z} \]
4. Step-by-step derivation

   1. lower-+.f64 67.9

      \[\leadsto \color{blue}{x + z} \]

5. Applied rewrites 67.9%

   \[\leadsto \color{blue}{x + z} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(FPCore (x y z)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, C"
  :precision binary64
  (+ (+ x (sin y)) (* z (cos y))))