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

Percentage Accurate: 99.9% → 99.9%

Time: 11.7s

Alternatives: 17

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

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

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]

Derivation

1. Initial program 99.9%

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

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ t_1 := z \cdot \sin y\\ t_2 := t\_0 - t\_1\\ t_3 := x - t\_1\\ \mathbf{if}\;t\_2 \leq -200000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y)))
        (t_1 (* z (sin y)))
        (t_2 (- t_0 t_1))
        (t_3 (- x t_1)))
   (if (<= t_2 -200000.0) t_3 (if (<= t_2 2000000.0) t_0 t_3))))

C

double code(double x, double y, double z) {
	double t_0 = x + cos(y);
	double t_1 = z * sin(y);
	double t_2 = t_0 - t_1;
	double t_3 = x - t_1;
	double tmp;
	if (t_2 <= -200000.0) {
		tmp = t_3;
	} else if (t_2 <= 2000000.0) {
		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 = x + cos(y)
    t_1 = z * sin(y)
    t_2 = t_0 - t_1
    t_3 = x - t_1
    if (t_2 <= (-200000.0d0)) then
        tmp = t_3
    else if (t_2 <= 2000000.0d0) 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 = x + Math.cos(y);
	double t_1 = z * Math.sin(y);
	double t_2 = t_0 - t_1;
	double t_3 = x - t_1;
	double tmp;
	if (t_2 <= -200000.0) {
		tmp = t_3;
	} else if (t_2 <= 2000000.0) {
		tmp = t_0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + math.cos(y)
	t_1 = z * math.sin(y)
	t_2 = t_0 - t_1
	t_3 = x - t_1
	tmp = 0
	if t_2 <= -200000.0:
		tmp = t_3
	elif t_2 <= 2000000.0:
		tmp = t_0
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + cos(y))
	t_1 = Float64(z * sin(y))
	t_2 = Float64(t_0 - t_1)
	t_3 = Float64(x - t_1)
	tmp = 0.0
	if (t_2 <= -200000.0)
		tmp = t_3;
	elseif (t_2 <= 2000000.0)
		tmp = t_0;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + cos(y);
	t_1 = z * sin(y);
	t_2 = t_0 - t_1;
	t_3 = x - t_1;
	tmp = 0.0;
	if (t_2 <= -200000.0)
		tmp = t_3;
	elseif (t_2 <= 2000000.0)
		tmp = t_0;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -200000.0], t$95$3, If[LessEqual[t$95$2, 2000000.0], 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))) < -2e5 or 2e6 < (-.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

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. associate-+l- N/A

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

      8. lower--.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-sin.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 99.1

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

   11. Applied rewrites 99.1%

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

   if -2e5 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 2e6

   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 97.5

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

   5. Applied rewrites 97.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -200000:\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{elif}\;\left(x + \cos y\right) - z \cdot \sin y \leq 2000000:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \cos y\right) - z \cdot \sin y\\ t_1 := \mathsf{fma}\left(\frac{y}{-1 + \frac{y \cdot \mathsf{fma}\left(\frac{y}{z}, -0.25, 0.5\right)}{z}}, z, x + 1\right)\\ \mathbf{if}\;t\_0 \leq -50:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.99999:\\ \;\;\;\;\cos y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (cos y)) (* z (sin y))))
        (t_1
         (fma (/ y (+ -1.0 (/ (* y (fma (/ y z) -0.25 0.5)) z))) z (+ x 1.0))))
   (if (<= t_0 -50.0) t_1 (if (<= t_0 0.99999) (cos y) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (x + cos(y)) - (z * sin(y));
	double t_1 = fma((y / (-1.0 + ((y * fma((y / z), -0.25, 0.5)) / z))), z, (x + 1.0));
	double tmp;
	if (t_0 <= -50.0) {
		tmp = t_1;
	} else if (t_0 <= 0.99999) {
		tmp = cos(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + cos(y)) - Float64(z * sin(y)))
	t_1 = fma(Float64(y / Float64(-1.0 + Float64(Float64(y * fma(Float64(y / z), -0.25, 0.5)) / z))), z, Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -50.0)
		tmp = t_1;
	elseif (t_0 <= 0.99999)
		tmp = cos(y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(-1.0 + N[(N[(y * N[(N[(y / z), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50.0], t$95$1, If[LessEqual[t$95$0, 0.99999], N[Cos[y], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -50 or 0.999990000000000046 < (-.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

      \[\leadsto \color{blue}{1 + \left(x + y \cdot \left(\frac{-1}{2} \cdot y - z\right)\right)} \]
   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. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 61.8

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

   5. Applied rewrites 61.8%

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

      1. lift-*.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 61.8

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

   7. Applied rewrites 61.8%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{1}{\color{blue}{y \cdot \left(\frac{-1}{4} \cdot \frac{y}{{z}^{3}} + \frac{1}{2} \cdot \frac{1}{{z}^{2}}\right) - \frac{1}{z}}}, x + 1\right) \]

   10. Applied rewrites 72.4%

       \[\leadsto \mathsf{fma}\left(y, \frac{1}{\color{blue}{\frac{-1 + \frac{\mathsf{fma}\left(y \cdot \frac{y}{z}, -0.25, y \cdot 0.5\right)}{z}}{z}}}, x + 1\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 72.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{-1 + \frac{y \cdot \mathsf{fma}\left(\frac{y}{z}, -0.25, 0.5\right)}{z}}, z, x + 1\right)} \]

   if -50 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.999990000000000046

   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 96.2

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in x around 0

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

      1. lower-cos.f64 93.6

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

   8. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\cos y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \mathsf{fma}\left(z, \sin y, -1\right)\\ \mathbf{if}\;x \leq -8 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-13}:\\ \;\;\;\;\cos y - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (fma z (sin y) -1.0))))
   (if (<= x -8e-12) t_0 (if (<= x 4e-13) (- (cos y) (* z (sin y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x - fma(z, sin(y), -1.0);
	double tmp;
	if (x <= -8e-12) {
		tmp = t_0;
	} else if (x <= 4e-13) {
		tmp = cos(y) - (z * sin(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x - fma(z, sin(y), -1.0))
	tmp = 0.0
	if (x <= -8e-12)
		tmp = t_0;
	elseif (x <= 4e-13)
		tmp = Float64(cos(y) - Float64(z * sin(y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(z * N[Sin[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e-12], t$95$0, If[LessEqual[x, 4e-13], N[(N[Cos[y], $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.99999999999999984e-12 or 4.0000000000000001e-13 < 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

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. associate-+l- N/A

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

      8. lower--.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-sin.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if -7.99999999999999984e-12 < x < 4.0000000000000001e-13

   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--.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 99.9

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

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\cos y - z \cdot \sin y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \mathsf{fma}\left(z, \sin y, -1\right)\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.000465:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (fma z (sin y) -1.0))))
   (if (<= z -1.6e+35) t_0 (if (<= z 0.000465) (+ x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x - fma(z, sin(y), -1.0);
	double tmp;
	if (z <= -1.6e+35) {
		tmp = t_0;
	} else if (z <= 0.000465) {
		tmp = x + cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x - fma(z, sin(y), -1.0))
	tmp = 0.0
	if (z <= -1.6e+35)
		tmp = t_0;
	elseif (z <= 0.000465)
		tmp = Float64(x + cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(z * N[Sin[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.6e+35], t$95$0, If[LessEqual[z, 0.000465], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.59999999999999991e35 or 4.6500000000000003e-4 < 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

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. associate-+l- N/A

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

      8. lower--.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-sin.f64 99.5

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

   8. Applied rewrites 99.5%

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

   if -1.59999999999999991e35 < z < 4.6500000000000003e-4

   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 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+35}:\\ \;\;\;\;x - \mathsf{fma}\left(z, \sin y, -1\right)\\ \mathbf{elif}\;z \leq 0.000465:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(z, \sin y, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+110}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) (- z))))
   (if (<= z -1.1e+159) t_0 (if (<= z 3.8e+110) (+ x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * -z;
	double tmp;
	if (z <= -1.1e+159) {
		tmp = t_0;
	} else if (z <= 3.8e+110) {
		tmp = x + cos(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 = sin(y) * -z
    if (z <= (-1.1d+159)) then
        tmp = t_0
    else if (z <= 3.8d+110) then
        tmp = x + cos(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 = Math.sin(y) * -z;
	double tmp;
	if (z <= -1.1e+159) {
		tmp = t_0;
	} else if (z <= 3.8e+110) {
		tmp = x + Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) * -z
	tmp = 0
	if z <= -1.1e+159:
		tmp = t_0
	elif z <= 3.8e+110:
		tmp = x + math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * Float64(-z))
	tmp = 0.0
	if (z <= -1.1e+159)
		tmp = t_0;
	elseif (z <= 3.8e+110)
		tmp = Float64(x + cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) * -z;
	tmp = 0.0;
	if (z <= -1.1e+159)
		tmp = t_0;
	elseif (z <= 3.8e+110)
		tmp = x + cos(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision]}, If[LessEqual[z, -1.1e+159], t$95$0, If[LessEqual[z, 3.8e+110], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1e159 or 3.79999999999999989e110 < z

   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. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-neg.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if -1.1e159 < z < 3.79999999999999989e110

   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 88.4

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

   5. Applied rewrites 88.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+159}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+110}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ \mathbf{if}\;y \leq -3200000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2600000000:\\ \;\;\;\;x - \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), z\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y))))
   (if (<= y -3200000000.0)
     t_0
     (if (<= y 2600000000.0)
       (-
        x
        (fma
         y
         (fma
          (* y y)
          (* z (fma (* y y) 0.008333333333333333 -0.16666666666666666))
          z)
         -1.0))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + cos(y);
	double tmp;
	if (y <= -3200000000.0) {
		tmp = t_0;
	} else if (y <= 2600000000.0) {
		tmp = x - fma(y, fma((y * y), (z * fma((y * y), 0.008333333333333333, -0.16666666666666666)), z), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + cos(y))
	tmp = 0.0
	if (y <= -3200000000.0)
		tmp = t_0;
	elseif (y <= 2600000000.0)
		tmp = Float64(x - fma(y, fma(Float64(y * y), Float64(z * fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666)), z), -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3200000000.0], t$95$0, If[LessEqual[y, 2600000000.0], N[(x - N[(y * N[(N[(y * y), $MachinePrecision] * N[(z * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.2e9 or 2.6e9 < 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 52.7

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

   5. Applied rewrites 52.7%

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

   if -3.2e9 < y < 2.6e9

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{1 \cdot x + \frac{1 - z \cdot \sin y}{x} \cdot x} \]

      4. *-lft-identity N/A

         \[\leadsto \color{blue}{x} + \frac{1 - z \cdot \sin y}{x} \cdot x \]

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

      12. distribute-lft-neg-out N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. associate-*r* N/A

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

   8. Applied rewrites 97.7%

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

   9. Taylor expanded in y around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. +-commutative N/A

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

       9. associate-*r* N/A

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

       10. distribute-rgt-out N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 98.6

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

   11. Applied rewrites 98.6%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3200000000:\\ \;\;\;\;x + \cos y\\ \mathbf{elif}\;y \leq 2600000000:\\ \;\;\;\;x - \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), z\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y}{-1 + \frac{y \cdot \mathsf{fma}\left(\frac{y}{z}, -0.25, 0.5\right)}{z}}, z, x + 1\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma (/ y (+ -1.0 (/ (* y (fma (/ y z) -0.25 0.5)) z))) z (+ x 1.0)))

C

double code(double x, double y, double z) {
	return fma((y / (-1.0 + ((y * fma((y / z), -0.25, 0.5)) / z))), z, (x + 1.0));
}

Julia

function code(x, y, z)
	return fma(Float64(y / Float64(-1.0 + Float64(Float64(y * fma(Float64(y / z), -0.25, 0.5)) / z))), z, Float64(x + 1.0))
end

Wolfram

code[x_, y_, z_] := N[(N[(y / N[(-1.0 + N[(N[(y * N[(N[(y / z), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + N[(x + 1.0), $MachinePrecision]), $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

   \[\leadsto \color{blue}{1 + \left(x + y \cdot \left(\frac{-1}{2} \cdot y - z\right)\right)} \]
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. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 55.1

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

5. Applied rewrites 55.1%

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

   1. lift-*.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 55.1

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

7. Applied rewrites 55.1%

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

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(y, \frac{1}{\color{blue}{y \cdot \left(\frac{-1}{4} \cdot \frac{y}{{z}^{3}} + \frac{1}{2} \cdot \frac{1}{{z}^{2}}\right) - \frac{1}{z}}}, x + 1\right) \]

10. Applied rewrites 65.5%

    \[\leadsto \mathsf{fma}\left(y, \frac{1}{\color{blue}{\frac{-1 + \frac{\mathsf{fma}\left(y \cdot \frac{y}{z}, -0.25, y \cdot 0.5\right)}{z}}{z}}}, x + 1\right) \]
11. Step-by-step derivation

12. Applied rewrites 65.5%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{-1 + \frac{y \cdot \mathsf{fma}\left(\frac{y}{z}, -0.25, 0.5\right)}{z}}, z, x + 1\right)} \]
13. Add Preprocessing

Alternative 9: 70.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+17}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 15500000000:\\ \;\;\;\;x - \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), z\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.35e+17)
   (+ x 1.0)
   (if (<= y 15500000000.0)
     (-
      x
      (fma
       y
       (fma
        (* y y)
        (* z (fma (* y y) 0.008333333333333333 -0.16666666666666666))
        z)
       -1.0))
     (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.35e+17) {
		tmp = x + 1.0;
	} else if (y <= 15500000000.0) {
		tmp = x - fma(y, fma((y * y), (z * fma((y * y), 0.008333333333333333, -0.16666666666666666)), z), -1.0);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.35e+17)
		tmp = Float64(x + 1.0);
	elseif (y <= 15500000000.0)
		tmp = Float64(x - fma(y, fma(Float64(y * y), Float64(z * fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666)), z), -1.0));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.35e+17], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 15500000000.0], N[(x - N[(y * N[(N[(y * y), $MachinePrecision] * N[(z * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35e17 or 1.55e10 < 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. +-commutative N/A

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

      2. lower-+.f64 33.0

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

   5. Applied rewrites 33.0%

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

   if -1.35e17 < y < 1.55e10

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{1 \cdot x + \frac{1 - z \cdot \sin y}{x} \cdot x} \]

      4. *-lft-identity N/A

         \[\leadsto \color{blue}{x} + \frac{1 - z \cdot \sin y}{x} \cdot x \]

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

      12. distribute-lft-neg-out N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. associate-*r* N/A

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

   8. Applied rewrites 97.0%

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

   9. Taylor expanded in y around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. +-commutative N/A

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

       9. associate-*r* N/A

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

       10. distribute-rgt-out N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 97.9

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

   11. Applied rewrites 97.9%

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

Alternative 10: 69.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+17}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+28}:\\ \;\;\;\;1 + \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, z \cdot 0.16666666666666666, -0.5\right) - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.05e+17)
   (+ x 1.0)
   (if (<= y 7.5e+28)
     (+ 1.0 (fma y (- (* y (fma y (* z 0.16666666666666666) -0.5)) z) x))
     (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.05e+17) {
		tmp = x + 1.0;
	} else if (y <= 7.5e+28) {
		tmp = 1.0 + fma(y, ((y * fma(y, (z * 0.16666666666666666), -0.5)) - z), x);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.05e+17)
		tmp = Float64(x + 1.0);
	elseif (y <= 7.5e+28)
		tmp = Float64(1.0 + fma(y, Float64(Float64(y * fma(y, Float64(z * 0.16666666666666666), -0.5)) - z), x));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.05e+17], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 7.5e+28], N[(1.0 + N[(y * N[(N[(y * N[(y * N[(z * 0.16666666666666666), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.05e17 or 7.4999999999999998e28 < 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. +-commutative N/A

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

      2. lower-+.f64 32.7

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

   5. Applied rewrites 32.7%

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

   if -2.05e17 < y < 7.4999999999999998e28

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 97.2

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

   5. Applied rewrites 97.2%

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

Alternative 11: 70.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.4:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 35000:\\ \;\;\;\;x + \mathsf{fma}\left(y, z \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6.4)
   (+ x 1.0)
   (if (<= y 35000.0)
     (+ x (fma y (* z (fma 0.16666666666666666 (* y y) -1.0)) 1.0))
     (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.4) {
		tmp = x + 1.0;
	} else if (y <= 35000.0) {
		tmp = x + fma(y, (z * fma(0.16666666666666666, (y * y), -1.0)), 1.0);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6.4)
		tmp = Float64(x + 1.0);
	elseif (y <= 35000.0)
		tmp = Float64(x + fma(y, Float64(z * fma(0.16666666666666666, Float64(y * y), -1.0)), 1.0));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6.4], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 35000.0], N[(x + N[(y * N[(z * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.4000000000000004 or 35000 < 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. +-commutative N/A

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

      2. lower-+.f64 33.3

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

   5. Applied rewrites 33.3%

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

   if -6.4000000000000004 < y < 35000

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. associate-+l- N/A

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

      8. lower--.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-sin.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. lower-+.f64 N/A

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

       4. lower-fma.f64 N/A

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

       5. sub-neg N/A

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

       6. associate-*r* N/A

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

       7. mul-1-neg N/A

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

       8. distribute-rgt-out N/A

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

       9. lower-*.f64 N/A

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

       10. lower-fma.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 12: 68.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+17}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.5 - z, x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.6e+17)
   (+ x 1.0)
   (if (<= y 5.8e+118) (fma y (- (* y -0.5) z) (+ x 1.0)) (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.6e+17) {
		tmp = x + 1.0;
	} else if (y <= 5.8e+118) {
		tmp = fma(y, ((y * -0.5) - z), (x + 1.0));
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.6e+17)
		tmp = Float64(x + 1.0);
	elseif (y <= 5.8e+118)
		tmp = fma(y, Float64(Float64(y * -0.5) - z), Float64(x + 1.0));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.6e+17], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 5.8e+118], N[(y * N[(N[(y * -0.5), $MachinePrecision] - z), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6e17 or 5.80000000000000032e118 < 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. +-commutative N/A

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

      2. lower-+.f64 34.7

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

   5. Applied rewrites 34.7%

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

   if -1.6e17 < y < 5.80000000000000032e118

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(x + y \cdot \left(\frac{-1}{2} \cdot y - z\right)\right)} \]
   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. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 88.5

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

   5. Applied rewrites 88.5%

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

Alternative 13: 69.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+78}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+118}:\\ \;\;\;\;x - \mathsf{fma}\left(y, z, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.9e+78)
   (+ x 1.0)
   (if (<= y 5.8e+118) (- x (fma y z -1.0)) (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.9e+78) {
		tmp = x + 1.0;
	} else if (y <= 5.8e+118) {
		tmp = x - fma(y, z, -1.0);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.9e+78)
		tmp = Float64(x + 1.0);
	elseif (y <= 5.8e+118)
		tmp = Float64(x - fma(y, z, -1.0));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.9e+78], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 5.8e+118], N[(x - N[(y * z + -1.0), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.90000000000000017e78 or 5.80000000000000032e118 < 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. +-commutative N/A

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

      2. lower-+.f64 38.0

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

   5. Applied rewrites 38.0%

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

   if -2.90000000000000017e78 < y < 5.80000000000000032e118

   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. metadata-eval N/A

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

      8. lower-fma.f64 81.2

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

   5. Applied rewrites 81.2%

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

Alternative 14: 66.4% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-11}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+22}:\\ \;\;\;\;1 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.12e-11) (+ x 1.0) (if (<= x 1.1e+22) (- 1.0 (* y z)) (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.12e-11) {
		tmp = x + 1.0;
	} else if (x <= 1.1e+22) {
		tmp = 1.0 - (y * z);
	} else {
		tmp = x + 1.0;
	}
	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) :: tmp
    if (x <= (-1.12d-11)) then
        tmp = x + 1.0d0
    else if (x <= 1.1d+22) then
        tmp = 1.0d0 - (y * z)
    else
        tmp = x + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.12e-11) {
		tmp = x + 1.0;
	} else if (x <= 1.1e+22) {
		tmp = 1.0 - (y * z);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.12e-11:
		tmp = x + 1.0
	elif x <= 1.1e+22:
		tmp = 1.0 - (y * z)
	else:
		tmp = x + 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.12e-11)
		tmp = Float64(x + 1.0);
	elseif (x <= 1.1e+22)
		tmp = Float64(1.0 - Float64(y * z));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.12e-11)
		tmp = x + 1.0;
	elseif (x <= 1.1e+22)
		tmp = 1.0 - (y * z);
	else
		tmp = x + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.12e-11], N[(x + 1.0), $MachinePrecision], If[LessEqual[x, 1.1e+22], N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1200000000000001e-11 or 1.1e22 < 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. +-commutative N/A

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

      2. lower-+.f64 77.9

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

   5. Applied rewrites 77.9%

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

   if -1.1200000000000001e-11 < x < 1.1e22

   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--.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{1 - y \cdot z} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - y \cdot z} \]

      4. *-commutative N/A

         \[\leadsto 1 - \color{blue}{z \cdot y} \]

      5. lower-*.f64 48.8

         \[\leadsto 1 - \color{blue}{z \cdot y} \]

   8. Applied rewrites 48.8%

      \[\leadsto \color{blue}{1 - z \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-11}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+22}:\\ \;\;\;\;1 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.7% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{+261}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 2.5e+261) (+ x 1.0) (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.5e+261) {
		tmp = x + 1.0;
	} 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 <= 2.5d+261) then
        tmp = x + 1.0d0
    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 <= 2.5e+261) {
		tmp = x + 1.0;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.5e+261:
		tmp = x + 1.0
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.5e+261)
		tmp = Float64(x + 1.0);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.5e+261)
		tmp = x + 1.0;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.5e+261], N[(x + 1.0), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.5e261

   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. +-commutative N/A

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

      2. lower-+.f64 59.8

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

   5. Applied rewrites 59.8%

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

   if 2.5e261 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(x + y \cdot \left(\frac{-1}{2} \cdot y - z\right)\right)} \]
   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. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 68.9

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 57.3

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

   8. Applied rewrites 57.3%

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

Alternative 16: 61.2% accurate, 53.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + 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 = x + 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x + 1.0), $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. +-commutative N/A

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

   2. lower-+.f64 57.4

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

5. Applied rewrites 57.4%

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

Alternative 17: 22.3% 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. +-commutative N/A

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

   2. lower-+.f64 57.4

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

5. Applied rewrites 57.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 21.1%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))))