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

Percentage Accurate: 99.9% → 99.9%

Time: 12.4s

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \mathsf{fma}\left(z, \sin y, \cos y\right)\\ t\_0 \cdot \frac{x + \left(\cos y - z \cdot \sin y\right)}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (fma z (sin y) (cos y)))))
   (* t_0 (/ (+ x (- (cos y) (* z (sin y)))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + fma(z, sin(y), cos(y));
	return t_0 * ((x + (cos(y) - (z * sin(y)))) / t_0);
}

Julia

function code(x, y, z)
	t_0 = Float64(x + fma(z, sin(y), cos(y)))
	return Float64(t_0 * Float64(Float64(x + Float64(cos(y) - Float64(z * sin(y)))) / t_0))
end

Wolfram

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

Derivation

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. difference-of-squares N/A

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

   4. lift--.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. associate-+l+ N/A

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

   9. lower-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(x + \mathsf{fma}\left(z, \sin y, \cos y\right)\right) \cdot \frac{x + \left(\cos y - z \cdot \sin y\right)}{x + \mathsf{fma}\left(z, \sin y, \cos y\right)}} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ t_1 := z \cdot \sin y\\ t_2 := t\_0 - t\_1\\ t_3 := x + \left(-t\_1\right)\\ \mathbf{if}\;t\_2 \leq -40000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 -40000000000.0) t_3 (if (<= t_2 5e+15) 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 <= -40000000000.0) {
		tmp = t_3;
	} else if (t_2 <= 5e+15) {
		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 <= (-40000000000.0d0)) then
        tmp = t_3
    else if (t_2 <= 5d+15) 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 <= -40000000000.0) {
		tmp = t_3;
	} else if (t_2 <= 5e+15) {
		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 <= -40000000000.0:
		tmp = t_3
	elif t_2 <= 5e+15:
		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 + Float64(-t_1))
	tmp = 0.0
	if (t_2 <= -40000000000.0)
		tmp = t_3;
	elseif (t_2 <= 5e+15)
		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 <= -40000000000.0)
		tmp = t_3;
	elseif (t_2 <= 5e+15)
		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, -40000000000.0], t$95$3, If[LessEqual[t$95$2, 5e+15], 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))) < -4e10 or 5e15 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y)))

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   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. lower-/.f64 99.6

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate--l+ N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 82.8

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

   7. Applied rewrites 82.8%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 99.6%

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

   if -4e10 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 5e15

   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 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -40000000000:\\ \;\;\;\;x + \left(-z \cdot \sin y\right)\\ \mathbf{elif}\;\left(x + \cos y\right) - z \cdot \sin y \leq 5 \cdot 10^{+15}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;x + \left(-z \cdot \sin y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 4: 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 5: 97.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+74}:\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{elif}\;z \leq 0.92:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8.8e+74)
   (- (+ x 1.0) (* z (sin y)))
   (if (<= z 0.92) (+ x (cos y)) (fma (sin y) (- z) (+ x 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.8e+74) {
		tmp = (x + 1.0) - (z * sin(y));
	} else if (z <= 0.92) {
		tmp = x + cos(y);
	} else {
		tmp = fma(sin(y), -z, (x + 1.0));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8.8e+74)
		tmp = Float64(Float64(x + 1.0) - Float64(z * sin(y)));
	elseif (z <= 0.92)
		tmp = Float64(x + cos(y));
	else
		tmp = fma(sin(y), Float64(-z), Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8.8e+74], N[(N[(x + 1.0), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.92], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * (-z) + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.8000000000000005e74

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.8%

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

   if -8.8000000000000005e74 < z < 0.92000000000000004

   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 99.5

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

   5. Applied rewrites 99.5%

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

   if 0.92000000000000004 < 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

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

   5. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-neg.f64 N/A

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

      8. lower-fma.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+74}:\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{elif}\;z \leq 0.92:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, x + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + 1\right) - z \cdot \sin y\\ \mathbf{if}\;z \leq -8.8 \cdot 10^{+74}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.92:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x 1.0) (* z (sin y)))))
   (if (<= z -8.8e+74) t_0 (if (<= z 0.92) (+ x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x + 1.0) - (z * sin(y));
	double tmp;
	if (z <= -8.8e+74) {
		tmp = t_0;
	} else if (z <= 0.92) {
		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 = (x + 1.0d0) - (z * sin(y))
    if (z <= (-8.8d+74)) then
        tmp = t_0
    else if (z <= 0.92d0) 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 = (x + 1.0) - (z * Math.sin(y));
	double tmp;
	if (z <= -8.8e+74) {
		tmp = t_0;
	} else if (z <= 0.92) {
		tmp = x + Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + 1.0) - (z * math.sin(y))
	tmp = 0
	if z <= -8.8e+74:
		tmp = t_0
	elif z <= 0.92:
		tmp = x + math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + 1.0) - Float64(z * sin(y)))
	tmp = 0.0
	if (z <= -8.8e+74)
		tmp = t_0;
	elseif (z <= 0.92)
		tmp = Float64(x + cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + 1.0) - (z * sin(y));
	tmp = 0.0;
	if (z <= -8.8e+74)
		tmp = t_0;
	elseif (z <= 0.92)
		tmp = x + cos(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8.8e+74], t$95$0, If[LessEqual[z, 0.92], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.8000000000000005e74 or 0.92000000000000004 < 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

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

   5. Applied rewrites 99.8%

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

   if -8.8000000000000005e74 < z < 0.92000000000000004

   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 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+74}:\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{elif}\;z \leq 0.92:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -z \cdot \sin y\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{+131}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+110}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* z (sin y)))))
   (if (<= z -2.2e+131) t_0 (if (<= z 3e+110) (+ x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -(z * sin(y));
	double tmp;
	if (z <= -2.2e+131) {
		tmp = t_0;
	} else if (z <= 3e+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 = -(z * sin(y))
    if (z <= (-2.2d+131)) then
        tmp = t_0
    else if (z <= 3d+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 = -(z * Math.sin(y));
	double tmp;
	if (z <= -2.2e+131) {
		tmp = t_0;
	} else if (z <= 3e+110) {
		tmp = x + Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -(z * math.sin(y))
	tmp = 0
	if z <= -2.2e+131:
		tmp = t_0
	elif z <= 3e+110:
		tmp = x + math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(z * sin(y)))
	tmp = 0.0
	if (z <= -2.2e+131)
		tmp = t_0;
	elseif (z <= 3e+110)
		tmp = Float64(x + cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -(z * sin(y));
	tmp = 0.0;
	if (z <= -2.2e+131)
		tmp = t_0;
	elseif (z <= 3e+110)
		tmp = x + cos(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[z, -2.2e+131], t$95$0, If[LessEqual[z, 3e+110], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1999999999999999e131 or 3.00000000000000007e110 < z

   1. Initial program 99.7%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   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. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 67.3

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

   5. Applied rewrites 67.3%

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

   if -2.1999999999999999e131 < z < 3.00000000000000007e110

   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 91.8

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

   5. Applied rewrites 91.8%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+131}:\\ \;\;\;\;-z \cdot \sin y\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+110}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;-z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ \mathbf{if}\;y \leq -0.03:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-22}:\\ \;\;\;\;\left(x + 1\right) - y \cdot \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y))))
   (if (<= y -0.03)
     t_0
     (if (<= y 3.6e-22)
       (-
        (+ x 1.0)
        (*
         y
         (fma
          (* y y)
          (* z (fma (* y y) 0.008333333333333333 -0.16666666666666666))
          z)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + cos(y);
	double tmp;
	if (y <= -0.03) {
		tmp = t_0;
	} else if (y <= 3.6e-22) {
		tmp = (x + 1.0) - (y * fma((y * y), (z * fma((y * y), 0.008333333333333333, -0.16666666666666666)), z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + cos(y))
	tmp = 0.0
	if (y <= -0.03)
		tmp = t_0;
	elseif (y <= 3.6e-22)
		tmp = Float64(Float64(x + 1.0) - Float64(y * fma(Float64(y * y), Float64(z * fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666)), z)));
	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, -0.03], t$95$0, If[LessEqual[y, 3.6e-22], N[(N[(x + 1.0), $MachinePrecision] - N[(y * N[(N[(y * y), $MachinePrecision] * N[(z * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.029999999999999999 or 3.5999999999999998e-22 < y

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   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 62.0

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

   5. Applied rewrites 62.0%

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

   if -0.029999999999999999 < y < 3.5999999999999998e-22

   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 100.0%

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

   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.03:\\ \;\;\;\;x + \cos y\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-22}:\\ \;\;\;\;\left(x + 1\right) - y \cdot \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+83}:\\ \;\;\;\;x + x \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+21}:\\ \;\;\;\;1 + \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot 0.16666666666666666, -0.5\right), -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8e+83)
   (+ x (* x (/ 1.0 x)))
   (if (<= y 4.1e+21)
     (+ 1.0 (fma y (fma 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 <= -8e+83) {
		tmp = x + (x * (1.0 / x));
	} else if (y <= 4.1e+21) {
		tmp = 1.0 + fma(y, fma(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 <= -8e+83)
		tmp = Float64(x + Float64(x * Float64(1.0 / x)));
	elseif (y <= 4.1e+21)
		tmp = Float64(1.0 + fma(y, fma(y, fma(y, Float64(z * 0.16666666666666666), -0.5), Float64(-z)), x));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8e+83], N[(x + N[(x * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.1e+21], N[(1.0 + N[(y * N[(y * N[(y * N[(z * 0.16666666666666666), $MachinePrecision] + -0.5), $MachinePrecision] + (-z)), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.00000000000000025e83

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   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. lower-/.f64 99.6

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate--l+ N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 81.2

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

   7. Applied rewrites 81.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 45.8%

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

   if -8.00000000000000025e83 < y < 4.1e21

   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. sub-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 91.5

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

   5. Applied rewrites 91.5%

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

   if 4.1e21 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 41.0

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

   5. Applied rewrites 41.0%

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

Alternative 10: 68.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+83}:\\ \;\;\;\;x + x \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;1 + \mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot \left(y \cdot 0.16666666666666666\right), -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8e+83)
   (+ x (* x (/ 1.0 x)))
   (if (<= y 3.6e+16)
     (+ 1.0 (fma y (fma y (* z (* y 0.16666666666666666)) (- z)) x))
     (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8e+83) {
		tmp = x + (x * (1.0 / x));
	} else if (y <= 3.6e+16) {
		tmp = 1.0 + fma(y, fma(y, (z * (y * 0.16666666666666666)), -z), x);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8e+83)
		tmp = Float64(x + Float64(x * Float64(1.0 / x)));
	elseif (y <= 3.6e+16)
		tmp = Float64(1.0 + fma(y, fma(y, Float64(z * Float64(y * 0.16666666666666666)), Float64(-z)), x));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8e+83], N[(x + N[(x * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.6e+16], N[(1.0 + N[(y * N[(y * N[(z * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + (-z)), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.00000000000000025e83

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   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. lower-/.f64 99.6

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate--l+ N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 81.2

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

   7. Applied rewrites 81.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 45.8%

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

   if -8.00000000000000025e83 < y < 3.6e16

   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. sub-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 92.1

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 92.0%

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

   if 3.6e16 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 40.2

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

   5. Applied rewrites 40.2%

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

Alternative 11: 69.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+64}:\\ \;\;\;\;x + x \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.5 - z, x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.5e+64)
   (+ x (* x (/ 1.0 x)))
   (if (<= y 4.1e+21) (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 <= -3.5e+64) {
		tmp = x + (x * (1.0 / x));
	} else if (y <= 4.1e+21) {
		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 <= -3.5e+64)
		tmp = Float64(x + Float64(x * Float64(1.0 / x)));
	elseif (y <= 4.1e+21)
		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, -3.5e+64], N[(x + N[(x * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.1e+21], 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 3 regimes

2. if y < -3.4999999999999999e64

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   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. lower-/.f64 99.6

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate--l+ N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 78.4

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

   7. Applied rewrites 78.4%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 42.5%

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

   if -3.4999999999999999e64 < y < 4.1e21

   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 94.4

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

   5. Applied rewrites 94.4%

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

   if 4.1e21 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 41.0

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

   5. Applied rewrites 41.0%

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

Alternative 12: 69.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+64}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.5 - z, x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.5e+64)
   (+ x 1.0)
   (if (<= y 4.1e+21) (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 <= -3.5e+64) {
		tmp = x + 1.0;
	} else if (y <= 4.1e+21) {
		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 <= -3.5e+64)
		tmp = Float64(x + 1.0);
	elseif (y <= 4.1e+21)
		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, -3.5e+64], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 4.1e+21], 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 < -3.4999999999999999e64 or 4.1e21 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 41.8

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

   5. Applied rewrites 41.8%

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

   if -3.4999999999999999e64 < y < 4.1e21

   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 94.4

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

   5. Applied rewrites 94.4%

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

Alternative 13: 68.8% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.1 \cdot 10^{+75}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-22}:\\ \;\;\;\;x - \mathsf{fma}\left(y, z, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.1e+75)
   (+ x 1.0)
   (if (<= y 3.6e-22) (- x (fma y z -1.0)) (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.1e+75) {
		tmp = x + 1.0;
	} else if (y <= 3.6e-22) {
		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 <= -7.1e+75)
		tmp = Float64(x + 1.0);
	elseif (y <= 3.6e-22)
		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, -7.1e+75], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 3.6e-22], N[(x - N[(y * z + -1.0), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.09999999999999982e75 or 3.5999999999999998e-22 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 43.9

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

   5. Applied rewrites 43.9%

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

   if -7.09999999999999982e75 < y < 3.5999999999999998e-22

   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 92.8

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

   5. Applied rewrites 92.8%

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

Alternative 14: 66.4% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-20}:\\ \;\;\;\;1 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -200.0) (+ x 1.0) (if (<= x 5e-20) (- 1.0 (* z y)) (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -200.0) {
		tmp = x + 1.0;
	} else if (x <= 5e-20) {
		tmp = 1.0 - (z * y);
	} 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 <= (-200.0d0)) then
        tmp = x + 1.0d0
    else if (x <= 5d-20) then
        tmp = 1.0d0 - (z * y)
    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 <= -200.0) {
		tmp = x + 1.0;
	} else if (x <= 5e-20) {
		tmp = 1.0 - (z * y);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -200.0:
		tmp = x + 1.0
	elif x <= 5e-20:
		tmp = 1.0 - (z * y)
	else:
		tmp = x + 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -200.0)
		tmp = Float64(x + 1.0);
	elseif (x <= 5e-20)
		tmp = Float64(1.0 - Float64(z * y));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -200.0)
		tmp = x + 1.0;
	elseif (x <= 5e-20)
		tmp = 1.0 - (z * y);
	else
		tmp = x + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -200.0], N[(x + 1.0), $MachinePrecision], If[LessEqual[x, 5e-20], N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -200 or 4.9999999999999999e-20 < 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 85.9

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

   5. Applied rewrites 85.9%

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

   if -200 < x < 4.9999999999999999e-20

   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. metadata-eval N/A

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

      8. lower-fma.f64 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.2%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-20}:\\ \;\;\;\;1 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.5% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 8.5 \cdot 10^{+206}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 8.5e+206) (+ x 1.0) (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 8.5e+206) {
		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 <= 8.5d+206) 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 <= 8.5e+206) {
		tmp = x + 1.0;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 8.5e+206:
		tmp = x + 1.0
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 8.5e+206)
		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 <= 8.5e+206)
		tmp = x + 1.0;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 8.5e+206], N[(x + 1.0), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8.4999999999999996e206

   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 67.9

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

   5. Applied rewrites 67.9%

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

   if 8.4999999999999996e206 < z

   1. Initial program 99.6%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   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 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 31.4%

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

Alternative 16: 61.0% 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 61.7

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

5. Applied rewrites 61.7%

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

Alternative 17: 20.9% 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 61.7

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

5. Applied rewrites 61.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 25.3%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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))))