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

Percentage Accurate: 99.9% → 99.9%

Time: 8.7s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + cos(y)) - (z * sin(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 + cos(y)) - (z * sin(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + cos(y)) - (z * sin(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 + cos(y)) - (z * sin(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (cos(y) + x) - (sin(y) * 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 = (cos(y) + x) - (sin(y) * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y + x\\ t_1 := \sin y \cdot z\\ t_2 := t\_0 - t\_1\\ t_3 := \left(1 + x\right) - t\_1\\ \mathbf{if}\;t\_2 \leq -1000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.9999931210694952:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (cos y) x))
        (t_1 (* (sin y) z))
        (t_2 (- t_0 t_1))
        (t_3 (- (+ 1.0 x) t_1)))
   (if (<= t_2 -1000000.0) t_3 (if (<= t_2 0.9999931210694952) t_0 t_3))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) + x;
	double t_1 = sin(y) * z;
	double t_2 = t_0 - t_1;
	double t_3 = (1.0 + x) - t_1;
	double tmp;
	if (t_2 <= -1000000.0) {
		tmp = t_3;
	} else if (t_2 <= 0.9999931210694952) {
		tmp = t_0;
	} else {
		tmp = t_3;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = cos(y) + x
    t_1 = sin(y) * z
    t_2 = t_0 - t_1
    t_3 = (1.0d0 + x) - t_1
    if (t_2 <= (-1000000.0d0)) then
        tmp = t_3
    else if (t_2 <= 0.9999931210694952d0) then
        tmp = t_0
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cos(y) + x;
	double t_1 = Math.sin(y) * z;
	double t_2 = t_0 - t_1;
	double t_3 = (1.0 + x) - t_1;
	double tmp;
	if (t_2 <= -1000000.0) {
		tmp = t_3;
	} else if (t_2 <= 0.9999931210694952) {
		tmp = t_0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cos(y) + x
	t_1 = math.sin(y) * z
	t_2 = t_0 - t_1
	t_3 = (1.0 + x) - t_1
	tmp = 0
	if t_2 <= -1000000.0:
		tmp = t_3
	elif t_2 <= 0.9999931210694952:
		tmp = t_0
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) + x)
	t_1 = Float64(sin(y) * z)
	t_2 = Float64(t_0 - t_1)
	t_3 = Float64(Float64(1.0 + x) - t_1)
	tmp = 0.0
	if (t_2 <= -1000000.0)
		tmp = t_3;
	elseif (t_2 <= 0.9999931210694952)
		tmp = t_0;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cos(y) + x;
	t_1 = sin(y) * z;
	t_2 = t_0 - t_1;
	t_3 = (1.0 + x) - t_1;
	tmp = 0.0;
	if (t_2 <= -1000000.0)
		tmp = t_3;
	elseif (t_2 <= 0.9999931210694952)
		tmp = t_0;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 + x), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -1000000.0], t$95$3, If[LessEqual[t$95$2, 0.9999931210694952], t$95$0, t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -1e6 or 0.99999312106949523 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.7%

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

   if -1e6 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.99999312106949523

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cos y + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos y + x\right) - \sin y \cdot z \leq -1000000:\\ \;\;\;\;\left(1 + x\right) - \sin y \cdot z\\ \mathbf{elif}\;\left(\cos y + x\right) - \sin y \cdot z \leq 0.9999931210694952:\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - \sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos y + x\right) - \sin y \cdot z\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 1 + x\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999931210694952:\\ \;\;\;\;\cos y\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (cos y) x) (* (sin y) z))))
   (if (<= t_0 -2e+23)
     (fma (- z) y (+ 1.0 x))
     (if (<= t_0 0.9999931210694952) (cos y) (- x (fma z y -1.0))))))

C

double code(double x, double y, double z) {
	double t_0 = (cos(y) + x) - (sin(y) * z);
	double tmp;
	if (t_0 <= -2e+23) {
		tmp = fma(-z, y, (1.0 + x));
	} else if (t_0 <= 0.9999931210694952) {
		tmp = cos(y);
	} else {
		tmp = x - fma(z, y, -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(cos(y) + x) - Float64(sin(y) * z))
	tmp = 0.0
	if (t_0 <= -2e+23)
		tmp = fma(Float64(-z), y, Float64(1.0 + x));
	elseif (t_0 <= 0.9999931210694952)
		tmp = cos(y);
	else
		tmp = Float64(x - fma(z, y, -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+23], N[((-z) * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999931210694952], N[Cos[y], $MachinePrecision], N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -1.9999999999999998e23

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.6

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-+.f64 65.1

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

   7. Applied rewrites 65.1%

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

   if -1.9999999999999998e23 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.99999312106949523

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 95.0

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

   7. Applied rewrites 95.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 88.5%

       \[\leadsto \cos y \]

   if 0.99999312106949523 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y)))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 78.6

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

   5. Applied rewrites 78.6%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos y + x\right) - \sin y \cdot z \leq -2 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 1 + x\right)\\ \mathbf{elif}\;\left(\cos y + x\right) - \sin y \cdot z \leq 0.9999931210694952:\\ \;\;\;\;\cos y\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot \sin y\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+141}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 265000000000:\\ \;\;\;\;\cos y + x\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+160}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - z, y, 1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) (sin y))))
   (if (<= z -1.7e+141)
     t_0
     (if (<= z 265000000000.0)
       (+ (cos y) x)
       (if (<= z 4e+160) t_0 (fma (- (* -0.5 y) z) y (+ 1.0 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = -z * sin(y);
	double tmp;
	if (z <= -1.7e+141) {
		tmp = t_0;
	} else if (z <= 265000000000.0) {
		tmp = cos(y) + x;
	} else if (z <= 4e+160) {
		tmp = t_0;
	} else {
		tmp = fma(((-0.5 * y) - z), y, (1.0 + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) * sin(y))
	tmp = 0.0
	if (z <= -1.7e+141)
		tmp = t_0;
	elseif (z <= 265000000000.0)
		tmp = Float64(cos(y) + x);
	elseif (z <= 4e+160)
		tmp = t_0;
	else
		tmp = fma(Float64(Float64(-0.5 * y) - z), y, Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.7e+141], t$95$0, If[LessEqual[z, 265000000000.0], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 4e+160], t$95$0, N[(N[(N[(-0.5 * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6999999999999999e141 or 2.65e11 < z < 4.00000000000000003e160

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 72.3

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

   5. Applied rewrites 72.3%

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

   if -1.6999999999999999e141 < z < 2.65e11

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 95.7

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

   5. Applied rewrites 95.7%

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

   if 4.00000000000000003e160 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 78.7

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

   5. Applied rewrites 78.7%

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

Alternative 5: 84.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \sin y \cdot z\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{+140}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 265000000000:\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (sin y) z))))
   (if (<= z -3.5e+140) t_0 (if (<= z 265000000000.0) (+ (cos y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (sin(y) * z);
	double tmp;
	if (z <= -3.5e+140) {
		tmp = t_0;
	} else if (z <= 265000000000.0) {
		tmp = cos(y) + x;
	} 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 = 1.0d0 - (sin(y) * z)
    if (z <= (-3.5d+140)) then
        tmp = t_0
    else if (z <= 265000000000.0d0) then
        tmp = cos(y) + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (Math.sin(y) * z);
	double tmp;
	if (z <= -3.5e+140) {
		tmp = t_0;
	} else if (z <= 265000000000.0) {
		tmp = Math.cos(y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (math.sin(y) * z)
	tmp = 0
	if z <= -3.5e+140:
		tmp = t_0
	elif z <= 265000000000.0:
		tmp = math.cos(y) + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(sin(y) * z))
	tmp = 0.0
	if (z <= -3.5e+140)
		tmp = t_0;
	elseif (z <= 265000000000.0)
		tmp = Float64(cos(y) + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (sin(y) * z);
	tmp = 0.0;
	if (z <= -3.5e+140)
		tmp = t_0;
	elseif (z <= 265000000000.0)
		tmp = cos(y) + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.5e+140], t$95$0, If[LessEqual[z, 265000000000.0], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.49999999999999989e140 or 2.65e11 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-cos.f64 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto 1 - z \cdot \sin y \]
   7. Step-by-step derivation

   8. Applied rewrites 75.7%

      \[\leadsto 1 - z \cdot \sin y \]

   if -3.49999999999999989e140 < z < 2.65e11

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 95.7

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

   5. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\cos y + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+140}:\\ \;\;\;\;1 - \sin y \cdot z\\ \mathbf{elif}\;z \leq 265000000000:\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;1 - \sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y + x\\ \mathbf{if}\;y \leq -10000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right) + x\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right) \cdot z, y \cdot y, z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (cos y) x)))
   (if (<= y -10000000.0)
     t_0
     (if (<= y 1.05)
       (-
        (+
         (fma
          (fma
           (fma -0.001388888888888889 (* y y) 0.041666666666666664)
           (* y y)
           -0.5)
          (* y y)
          1.0)
         x)
        (*
         (fma
          (* (fma 0.008333333333333333 (* y y) -0.16666666666666666) z)
          (* y y)
          z)
         y))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) + x;
	double tmp;
	if (y <= -10000000.0) {
		tmp = t_0;
	} else if (y <= 1.05) {
		tmp = (fma(fma(fma(-0.001388888888888889, (y * y), 0.041666666666666664), (y * y), -0.5), (y * y), 1.0) + x) - (fma((fma(0.008333333333333333, (y * y), -0.16666666666666666) * z), (y * y), z) * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) + x)
	tmp = 0.0
	if (y <= -10000000.0)
		tmp = t_0;
	elseif (y <= 1.05)
		tmp = Float64(Float64(fma(fma(fma(-0.001388888888888889, Float64(y * y), 0.041666666666666664), Float64(y * y), -0.5), Float64(y * y), 1.0) + x) - Float64(fma(Float64(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666) * z), Float64(y * y), z) * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -10000000.0], t$95$0, If[LessEqual[y, 1.05], N[(N[(N[(N[(N[(-0.001388888888888889 * N[(y * y), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision] - N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * z), $MachinePrecision] * N[(y * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1e7 or 1.05000000000000004 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 62.8

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

   5. Applied rewrites 62.8%

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

   if -1e7 < y < 1.05000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 97.9%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(x + \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)}\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(x + \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) + 1\right)}\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       2. *-commutative N/A

          \[\leadsto \left(x + \left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) \cdot {y}^{2}} + 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(x + \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, {y}^{2}, 1\right)}\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       4. sub-neg N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       5. *-commutative N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       6. metadata-eval N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2} + \color{blue}{\frac{-1}{2}}, {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, {y}^{2}, \frac{-1}{2}\right)}, {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       8. +-commutative N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{720} \cdot {y}^{2} + \frac{1}{24}}, {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       9. lower-fma.f64 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{720}, {y}^{2}, \frac{1}{24}\right)}, {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       10. unpow2 N/A

           \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{y \cdot y}, \frac{1}{24}\right), {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       11. lower-*.f64 N/A

           \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{y \cdot y}, \frac{1}{24}\right), {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       12. unpow2 N/A

           \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       13. lower-*.f64 N/A

           \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       14. unpow2 N/A

           \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), y \cdot y, \frac{-1}{2}\right), \color{blue}{y \cdot y}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       15. lower-*.f64 98.6

           \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), \color{blue}{y \cdot y}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y \]

   11. Applied rewrites 98.6%

       \[\leadsto \left(x + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right)}\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10000000:\\ \;\;\;\;\cos y + x\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right) + x\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right) \cdot z, y \cdot y, z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\cos y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -10000000:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 1350000:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right) + x\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right) \cdot z, y \cdot y, z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -10000000.0)
   (+ 1.0 x)
   (if (<= y 1350000.0)
     (-
      (+
       (fma
        (fma
         (fma -0.001388888888888889 (* y y) 0.041666666666666664)
         (* y y)
         -0.5)
        (* y y)
        1.0)
       x)
      (*
       (fma
        (* (fma 0.008333333333333333 (* y y) -0.16666666666666666) z)
        (* y y)
        z)
       y))
     (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -10000000.0) {
		tmp = 1.0 + x;
	} else if (y <= 1350000.0) {
		tmp = (fma(fma(fma(-0.001388888888888889, (y * y), 0.041666666666666664), (y * y), -0.5), (y * y), 1.0) + x) - (fma((fma(0.008333333333333333, (y * y), -0.16666666666666666) * z), (y * y), z) * y);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -10000000.0)
		tmp = Float64(1.0 + x);
	elseif (y <= 1350000.0)
		tmp = Float64(Float64(fma(fma(fma(-0.001388888888888889, Float64(y * y), 0.041666666666666664), Float64(y * y), -0.5), Float64(y * y), 1.0) + x) - Float64(fma(Float64(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666) * z), Float64(y * y), z) * y));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -10000000.0], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 1350000.0], N[(N[(N[(N[(N[(-0.001388888888888889 * N[(y * y), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision] - N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * z), $MachinePrecision] * N[(y * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1e7 or 1.35e6 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. lower-+.f64 40.1

         \[\leadsto \color{blue}{1 + x} \]

   5. Applied rewrites 40.1%

      \[\leadsto \color{blue}{1 + x} \]

   if -1e7 < y < 1.35e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 95.8%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(x + \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)}\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(x + \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) + 1\right)}\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       2. *-commutative N/A

          \[\leadsto \left(x + \left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) \cdot {y}^{2}} + 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(x + \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, {y}^{2}, 1\right)}\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       4. sub-neg N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       5. *-commutative N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       6. metadata-eval N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2} + \color{blue}{\frac{-1}{2}}, {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, {y}^{2}, \frac{-1}{2}\right)}, {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       8. +-commutative N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{720} \cdot {y}^{2} + \frac{1}{24}}, {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       9. lower-fma.f64 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{720}, {y}^{2}, \frac{1}{24}\right)}, {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       10. unpow2 N/A

           \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{y \cdot y}, \frac{1}{24}\right), {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       11. lower-*.f64 N/A

           \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{y \cdot y}, \frac{1}{24}\right), {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       12. unpow2 N/A

           \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       13. lower-*.f64 N/A

           \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       14. unpow2 N/A

           \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), y \cdot y, \frac{-1}{2}\right), \color{blue}{y \cdot y}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       15. lower-*.f64 96.5

           \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), \color{blue}{y \cdot y}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y \]

   11. Applied rewrites 96.5%

       \[\leadsto \left(x + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right)}\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10000000:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 1350000:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right) + x\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right) \cdot z, y \cdot y, z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -10000000:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 3.2:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot y, -0.5\right), y \cdot y, 1\right) + x\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right) \cdot z, y \cdot y, z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -10000000.0)
   (+ 1.0 x)
   (if (<= y 3.2)
     (-
      (+ (fma (fma 0.041666666666666664 (* y y) -0.5) (* y y) 1.0) x)
      (*
       (fma
        (* (fma 0.008333333333333333 (* y y) -0.16666666666666666) z)
        (* y y)
        z)
       y))
     (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -10000000.0) {
		tmp = 1.0 + x;
	} else if (y <= 3.2) {
		tmp = (fma(fma(0.041666666666666664, (y * y), -0.5), (y * y), 1.0) + x) - (fma((fma(0.008333333333333333, (y * y), -0.16666666666666666) * z), (y * y), z) * y);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -10000000.0)
		tmp = Float64(1.0 + x);
	elseif (y <= 3.2)
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(y * y), -0.5), Float64(y * y), 1.0) + x) - Float64(fma(Float64(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666) * z), Float64(y * y), z) * y));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -10000000.0], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 3.2], N[(N[(N[(N[(0.041666666666666664 * N[(y * y), $MachinePrecision] + -0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision] - N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * z), $MachinePrecision] * N[(y * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1e7 or 3.2000000000000002 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. lower-+.f64 39.1

         \[\leadsto \color{blue}{1 + x} \]

   5. Applied rewrites 39.1%

      \[\leadsto \color{blue}{1 + x} \]

   if -1e7 < y < 3.2000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 97.9%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(x + \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)\right)}\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right) + 1\right)}\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       2. *-commutative N/A

          \[\leadsto \left(x + \left(\color{blue}{\left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right) \cdot {y}^{2}} + 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(x + \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)}\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       4. sub-neg N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       5. metadata-eval N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot {y}^{2} + \color{blue}{\frac{-1}{2}}, {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {y}^{2}, \frac{-1}{2}\right)}, {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       7. unpow2 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       8. lower-*.f64 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       9. unpow2 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, y \cdot y, \frac{-1}{2}\right), \color{blue}{y \cdot y}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, z\right) \cdot y \]

       10. lower-*.f64 98.5

           \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot y, -0.5\right), \color{blue}{y \cdot y}, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y \]

   11. Applied rewrites 98.5%

       \[\leadsto \left(x + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot y, -0.5\right), y \cdot y, 1\right)}\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10000000:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 3.2:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot y, -0.5\right), y \cdot y, 1\right) + x\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right) \cdot z, y \cdot y, z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.9% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - z \cdot y\\ \mathbf{if}\;z \leq -2.55 \cdot 10^{+148}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.38 \cdot 10^{+166}:\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (* z y))))
   (if (<= z -2.55e+148) t_0 (if (<= z 1.38e+166) (+ 1.0 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x - (z * y);
	double tmp;
	if (z <= -2.55e+148) {
		tmp = t_0;
	} else if (z <= 1.38e+166) {
		tmp = 1.0 + x;
	} 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 = x - (z * y)
    if (z <= (-2.55d+148)) then
        tmp = t_0
    else if (z <= 1.38d+166) then
        tmp = 1.0d0 + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x - (z * y);
	double tmp;
	if (z <= -2.55e+148) {
		tmp = t_0;
	} else if (z <= 1.38e+166) {
		tmp = 1.0 + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - (z * y)
	tmp = 0
	if z <= -2.55e+148:
		tmp = t_0
	elif z <= 1.38e+166:
		tmp = 1.0 + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(z * y))
	tmp = 0.0
	if (z <= -2.55e+148)
		tmp = t_0;
	elseif (z <= 1.38e+166)
		tmp = Float64(1.0 + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x - (z * y);
	tmp = 0.0;
	if (z <= -2.55e+148)
		tmp = t_0;
	elseif (z <= 1.38e+166)
		tmp = 1.0 + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.55e+148], t$95$0, If[LessEqual[z, 1.38e+166], N[(1.0 + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.54999999999999993e148 or 1.38000000000000001e166 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 60.4%

      \[\leadsto x - z \cdot \color{blue}{y} \]

   if -2.54999999999999993e148 < z < 1.38000000000000001e166

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. lower-+.f64 69.3

         \[\leadsto \color{blue}{1 + x} \]

   5. Applied rewrites 69.3%

      \[\leadsto \color{blue}{1 + x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 66.0% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.18 \cdot 10^{+23}:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.18e+23)
   (+ 1.0 x)
   (if (<= x 9.2e-7) (fma (- y) z 1.0) (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.18e+23) {
		tmp = 1.0 + x;
	} else if (x <= 9.2e-7) {
		tmp = fma(-y, z, 1.0);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.18e+23)
		tmp = Float64(1.0 + x);
	elseif (x <= 9.2e-7)
		tmp = fma(Float64(-y), z, 1.0);
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.18e+23], N[(1.0 + x), $MachinePrecision], If[LessEqual[x, 9.2e-7], N[((-y) * z + 1.0), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.18e23 or 9.1999999999999998e-7 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. lower-+.f64 81.9

         \[\leadsto \color{blue}{1 + x} \]

   5. Applied rewrites 81.9%

      \[\leadsto \color{blue}{1 + x} \]

   if -1.18e23 < x < 9.1999999999999998e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 51.6

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{y \cdot z} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.1%

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

Alternative 11: 66.2% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4600000000:\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4600000000.0) (+ 1.0 x) (fma (- z) y (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4600000000.0) {
		tmp = 1.0 + x;
	} else {
		tmp = fma(-z, y, (1.0 + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4600000000.0)
		tmp = Float64(1.0 + x);
	else
		tmp = fma(Float64(-z), y, Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4600000000.0], N[(1.0 + x), $MachinePrecision], N[((-z) * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.6e9

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. lower-+.f64 44.2

         \[\leadsto \color{blue}{1 + x} \]

   5. Applied rewrites 44.2%

      \[\leadsto \color{blue}{1 + x} \]

   if -4.6e9 < y

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-+.f64 76.7

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

   7. Applied rewrites 76.7%

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

Alternative 12: 66.2% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4600000000:\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4600000000.0) (+ 1.0 x) (- x (fma z y -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4600000000.0) {
		tmp = 1.0 + x;
	} else {
		tmp = x - fma(z, y, -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4600000000.0)
		tmp = Float64(1.0 + x);
	else
		tmp = Float64(x - fma(z, y, -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4600000000.0], N[(1.0 + x), $MachinePrecision], N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.6e9

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. lower-+.f64 44.2

         \[\leadsto \color{blue}{1 + x} \]

   5. Applied rewrites 44.2%

      \[\leadsto \color{blue}{1 + x} \]

   if -4.6e9 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 76.7

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

   5. Applied rewrites 76.7%

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

Alternative 13: 62.5% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{+255}:\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 2.3e+255) (+ 1.0 x) (* (- y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.3e+255) {
		tmp = 1.0 + x;
	} else {
		tmp = -y * 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) :: tmp
    if (z <= 2.3d+255) then
        tmp = 1.0d0 + x
    else
        tmp = -y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.3e+255) {
		tmp = 1.0 + x;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.3e+255:
		tmp = 1.0 + x
	else:
		tmp = -y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.3e+255)
		tmp = Float64(1.0 + x);
	else
		tmp = Float64(Float64(-y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.3e+255)
		tmp = 1.0 + x;
	else
		tmp = -y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.3e+255], N[(1.0 + x), $MachinePrecision], N[((-y) * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.3e255

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. lower-+.f64 63.3

         \[\leadsto \color{blue}{1 + x} \]

   5. Applied rewrites 63.3%

      \[\leadsto \color{blue}{1 + x} \]

   if 2.3e255 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 65.1%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{+255}:\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.8% accurate, 53.0× speedup

Math

\[\begin{array}{l} \\ 1 + x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ 1.0 x))

C

double code(double x, double y, double z) {
	return 1.0 + x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + x
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + x;
}

Python

def code(x, y, z):
	return 1.0 + x

Julia

function code(x, y, z)
	return Float64(1.0 + x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + x;
end

Wolfram

code[x_, y_, z_] := N[(1.0 + x), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1 + x} \]
4. Step-by-step derivation

   1. lower-+.f64 59.2

      \[\leadsto \color{blue}{1 + x} \]

5. Applied rewrites 59.2%

   \[\leadsto \color{blue}{1 + x} \]
6. Add Preprocessing

Alternative 15: 20.8% accurate, 212.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 1.0)

C

double code(double x, double y, double z) {
	return 1.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 1.0;
}

Python

def code(x, y, z):
	return 1.0

Julia

function code(x, y, z)
	return 1.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1 + x} \]
4. Step-by-step derivation

   1. lower-+.f64 59.2

      \[\leadsto \color{blue}{1 + x} \]

5. Applied rewrites 59.2%

   \[\leadsto \color{blue}{1 + x} \]

6. Taylor expanded in x around 0

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 21.2%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024254 
(FPCore (x y z)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, B"
  :precision binary64
  (- (+ x (cos y)) (* z (sin y))))