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

Percentage Accurate: 99.9% → 99.9%

Time: 5.3s

Alternatives: 12

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(x + N[((-z) * N[Sin[y], $MachinePrecision] + 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. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. lower-fma.f64 N/A

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

   11. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot z\\ t_1 := \left(x + \cos y\right) - t\_0\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\left(-\sin y\right) \cdot z + x\\ \mathbf{elif}\;t\_1 \leq 0.998:\\ \;\;\;\;\cos y - t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) z)) (t_1 (- (+ x (cos y)) t_0)))
   (if (<= t_1 -2e+27)
     (+ (* (- (sin y)) z) x)
     (if (<= t_1 0.998) (- (cos y) t_0) (- (+ 1.0 x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * z;
	double t_1 = (x + cos(y)) - t_0;
	double tmp;
	if (t_1 <= -2e+27) {
		tmp = (-sin(y) * z) + x;
	} else if (t_1 <= 0.998) {
		tmp = cos(y) - t_0;
	} else {
		tmp = (1.0 + x) - 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) :: t_1
    real(8) :: tmp
    t_0 = sin(y) * z
    t_1 = (x + cos(y)) - t_0
    if (t_1 <= (-2d+27)) then
        tmp = (-sin(y) * z) + x
    else if (t_1 <= 0.998d0) then
        tmp = cos(y) - t_0
    else
        tmp = (1.0d0 + x) - 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 t_1 = (x + Math.cos(y)) - t_0;
	double tmp;
	if (t_1 <= -2e+27) {
		tmp = (-Math.sin(y) * z) + x;
	} else if (t_1 <= 0.998) {
		tmp = Math.cos(y) - t_0;
	} else {
		tmp = (1.0 + x) - t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) * z
	t_1 = (x + math.cos(y)) - t_0
	tmp = 0
	if t_1 <= -2e+27:
		tmp = (-math.sin(y) * z) + x
	elif t_1 <= 0.998:
		tmp = math.cos(y) - t_0
	else:
		tmp = (1.0 + x) - t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * z)
	t_1 = Float64(Float64(x + cos(y)) - t_0)
	tmp = 0.0
	if (t_1 <= -2e+27)
		tmp = Float64(Float64(Float64(-sin(y)) * z) + x);
	elseif (t_1 <= 0.998)
		tmp = Float64(cos(y) - t_0);
	else
		tmp = Float64(Float64(1.0 + x) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) * z;
	t_1 = (x + cos(y)) - t_0;
	tmp = 0.0;
	if (t_1 <= -2e+27)
		tmp = (-sin(y) * z) + x;
	elseif (t_1 <= 0.998)
		tmp = cos(y) - t_0;
	else
		tmp = (1.0 + x) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+27], N[(N[((-N[Sin[y], $MachinePrecision]) * z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 0.998], N[(N[Cos[y], $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 + x), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -2e27

   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. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 99.9%

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

   if -2e27 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.998

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-cos.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 0.998 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - \sin y \cdot z \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\left(-\sin y\right) \cdot z + x\\ \mathbf{elif}\;\left(x + \cos y\right) - \sin y \cdot z \leq 0.998:\\ \;\;\;\;\cos y - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - \sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \cos y\right) - \sin y \cdot z\\ t_1 := \left(-\sin y\right) \cdot z + x\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\cos y - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (cos y)) (* (sin y) z))) (t_1 (+ (* (- (sin y)) z) x)))
   (if (<= t_0 -5e+19) t_1 (if (<= t_0 2.0) (- (cos y) (* y z)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (x + cos(y)) - (sin(y) * z);
	double t_1 = (-sin(y) * z) + x;
	double tmp;
	if (t_0 <= -5e+19) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = cos(y) - (y * z);
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_0 = (x + cos(y)) - (sin(y) * z)
    t_1 = (-sin(y) * z) + x
    if (t_0 <= (-5d+19)) then
        tmp = t_1
    else if (t_0 <= 2.0d0) then
        tmp = cos(y) - (y * z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + Math.cos(y)) - (Math.sin(y) * z);
	double t_1 = (-Math.sin(y) * z) + x;
	double tmp;
	if (t_0 <= -5e+19) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = Math.cos(y) - (y * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + math.cos(y)) - (math.sin(y) * z)
	t_1 = (-math.sin(y) * z) + x
	tmp = 0
	if t_0 <= -5e+19:
		tmp = t_1
	elif t_0 <= 2.0:
		tmp = math.cos(y) - (y * z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + cos(y)) - Float64(sin(y) * z))
	t_1 = Float64(Float64(Float64(-sin(y)) * z) + x)
	tmp = 0.0
	if (t_0 <= -5e+19)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(cos(y) - Float64(y * z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + cos(y)) - (sin(y) * z);
	t_1 = (-sin(y) * z) + x;
	tmp = 0.0;
	if (t_0 <= -5e+19)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = cos(y) - (y * z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-N[Sin[y], $MachinePrecision]) * z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+19], t$95$1, If[LessEqual[t$95$0, 2.0], N[(N[Cos[y], $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -5e19 or 2 < (-.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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 98.4%

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

   if -5e19 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 83.5

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

   5. Applied rewrites 83.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-cos.f64 83.5

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

   8. Applied rewrites 83.5%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - \sin y \cdot z \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\left(-\sin y\right) \cdot z + x\\ \mathbf{elif}\;\left(x + \cos y\right) - \sin y \cdot z \leq 2:\\ \;\;\;\;\cos y - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin y\right) \cdot z + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + x\right) - \sin y \cdot z\\ \mathbf{if}\;z \leq -2.25 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.36:\\ \;\;\;\;\frac{\cos y}{z} \cdot z + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ 1.0 x) (* (sin y) z))))
   (if (<= z -2.25e+18) t_0 (if (<= z 0.36) (+ (* (/ (cos y) z) z) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (1.0 + x) - (sin(y) * z);
	double tmp;
	if (z <= -2.25e+18) {
		tmp = t_0;
	} else if (z <= 0.36) {
		tmp = ((cos(y) / z) * z) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + x) - (sin(y) * z)
    if (z <= (-2.25d+18)) then
        tmp = t_0
    else if (z <= 0.36d0) then
        tmp = ((cos(y) / z) * z) + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (1.0 + x) - (Math.sin(y) * z);
	double tmp;
	if (z <= -2.25e+18) {
		tmp = t_0;
	} else if (z <= 0.36) {
		tmp = ((Math.cos(y) / z) * z) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (1.0 + x) - (math.sin(y) * z)
	tmp = 0
	if z <= -2.25e+18:
		tmp = t_0
	elif z <= 0.36:
		tmp = ((math.cos(y) / z) * z) + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(1.0 + x) - Float64(sin(y) * z))
	tmp = 0.0
	if (z <= -2.25e+18)
		tmp = t_0;
	elseif (z <= 0.36)
		tmp = Float64(Float64(Float64(cos(y) / z) * z) + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (1.0 + x) - (sin(y) * z);
	tmp = 0.0;
	if (z <= -2.25e+18)
		tmp = t_0;
	elseif (z <= 0.36)
		tmp = ((cos(y) / z) * z) + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 + x), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.25e+18], t$95$0, If[LessEqual[z, 0.36], N[(N[(N[(N[Cos[y], $MachinePrecision] / z), $MachinePrecision] * z), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.25e18 or 0.35999999999999999 < 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 -2.25e18 < z < 0.35999999999999999

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 81.4%

       \[\leadsto \frac{1}{z} \cdot z + x \]

   11. Taylor expanded in z around 0

       \[\leadsto \frac{\cos y}{z} \cdot z + x \]
   12. Step-by-step derivation

   13. Applied rewrites 99.0%

       \[\leadsto \frac{\cos y}{z} \cdot z + x \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+18}:\\ \;\;\;\;\left(1 + x\right) - \sin y \cdot z\\ \mathbf{elif}\;z \leq 0.36:\\ \;\;\;\;\frac{\cos y}{z} \cdot z + x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - \sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-\sin y\right) \cdot z\\ \mathbf{if}\;y \leq -7000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y \cdot z, -0.5\right) \cdot y - z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (sin y)) z)))
   (if (<= y -7000000.0)
     t_0
     (if (<= y 7.4)
       (fma (- (* (fma 0.16666666666666666 (* y z) -0.5) y) z) y (+ 1.0 x))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -sin(y) * z;
	double tmp;
	if (y <= -7000000.0) {
		tmp = t_0;
	} else if (y <= 7.4) {
		tmp = fma(((fma(0.16666666666666666, (y * z), -0.5) * y) - z), y, (1.0 + x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-sin(y)) * z)
	tmp = 0.0
	if (y <= -7000000.0)
		tmp = t_0;
	elseif (y <= 7.4)
		tmp = fma(Float64(Float64(fma(0.16666666666666666, Float64(y * z), -0.5) * y) - z), y, Float64(1.0 + x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-N[Sin[y], $MachinePrecision]) * z), $MachinePrecision]}, If[LessEqual[y, -7000000.0], t$95$0, If[LessEqual[y, 7.4], N[(N[(N[(N[(0.16666666666666666 * N[(y * z), $MachinePrecision] + -0.5), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7e6 or 7.4000000000000004 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 50.2

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

   5. Applied rewrites 50.2%

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

   if -7e6 < y < 7.4000000000000004

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7000000:\\ \;\;\;\;\left(-\sin y\right) \cdot z\\ \mathbf{elif}\;y \leq 7.4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y \cdot z, -0.5\right) \cdot y - z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(1 + x\right) - \sin y \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ 1.0 x) (* (sin y) z)))

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	return (1.0 + x) - (Math.sin(y) * z);
}

Python

def code(x, y, z):
	return (1.0 + x) - (math.sin(y) * z)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. 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 91.3%

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

6. Final simplification 91.3%

   \[\leadsto \left(1 + x\right) - \sin y \cdot z \]
7. Add Preprocessing

Alternative 7: 69.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -230000:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 530000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y \cdot z, -0.5\right) \cdot y - z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -230000.0)
   (+ 1.0 x)
   (if (<= y 530000.0)
     (fma (- (* (fma 0.16666666666666666 (* y z) -0.5) y) z) y (+ 1.0 x))
     (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -230000.0) {
		tmp = 1.0 + x;
	} else if (y <= 530000.0) {
		tmp = fma(((fma(0.16666666666666666, (y * z), -0.5) * y) - z), y, (1.0 + x));
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -230000.0)
		tmp = Float64(1.0 + x);
	elseif (y <= 530000.0)
		tmp = fma(Float64(Float64(fma(0.16666666666666666, Float64(y * z), -0.5) * y) - z), y, Float64(1.0 + x));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -230000.0], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 530000.0], N[(N[(N[(N[(0.16666666666666666 * N[(y * z), $MachinePrecision] + -0.5), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.3e5 or 5.3e5 < 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. lower-+.f64 33.3

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

   5. Applied rewrites 33.3%

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

   if -2.3e5 < y < 5.3e5

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -230000:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 530000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y \cdot z, -0.5\right) \cdot y - z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.04 \cdot 10^{+104}:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.04e+104)
   (+ 1.0 x)
   (if (<= y 5.5e+17) (fma (- (* -0.5 y) z) y (+ 1.0 x)) (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.04e+104) {
		tmp = 1.0 + x;
	} else if (y <= 5.5e+17) {
		tmp = fma(((-0.5 * y) - z), y, (1.0 + x));
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.04e+104)
		tmp = Float64(1.0 + x);
	elseif (y <= 5.5e+17)
		tmp = fma(Float64(Float64(-0.5 * y) - z), y, Float64(1.0 + x));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.04e+104], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 5.5e+17], N[(N[(N[(-0.5 * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.0400000000000001e104 or 5.5e17 < 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. lower-+.f64 34.1

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

   5. Applied rewrites 34.1%

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

   if -1.0400000000000001e104 < y < 5.5e17

   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 92.1

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

   5. Applied rewrites 92.1%

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

Alternative 9: 69.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3350000000000:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+69}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3350000000000.0)
   (+ 1.0 x)
   (if (<= y 1.8e+69) (- x (fma z y -1.0)) (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3350000000000.0) {
		tmp = 1.0 + x;
	} else if (y <= 1.8e+69) {
		tmp = x - fma(z, y, -1.0);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3350000000000.0)
		tmp = Float64(1.0 + x);
	elseif (y <= 1.8e+69)
		tmp = Float64(x - fma(z, y, -1.0));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3350000000000.0], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 1.8e+69], N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.35e12 or 1.8000000000000001e69 < 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. lower-+.f64 33.5

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

   5. Applied rewrites 33.5%

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

   if -3.35e12 < y < 1.8000000000000001e69

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 95.3

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

   5. Applied rewrites 95.3%

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

Alternative 10: 66.7% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{-10}:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.08e-10)
   (+ 1.0 x)
   (if (<= x 1.8e-25) (fma (- z) y 1.0) (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.08e-10) {
		tmp = 1.0 + x;
	} else if (x <= 1.8e-25) {
		tmp = fma(-z, y, 1.0);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.08e-10)
		tmp = Float64(1.0 + x);
	elseif (x <= 1.8e-25)
		tmp = fma(Float64(-z), y, 1.0);
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.08e-10], N[(1.0 + x), $MachinePrecision], If[LessEqual[x, 1.8e-25], N[((-z) * y + 1.0), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.08000000000000002e-10 or 1.8e-25 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 77.0

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

   5. Applied rewrites 77.0%

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

   if -1.08000000000000002e-10 < x < 1.8e-25

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 57.5

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

   5. Applied rewrites 57.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, z \cdot y, \frac{-1}{2}\right) \cdot y - z, y, 1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 57.5%

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

   9. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot z, y, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 58.3%

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

Alternative 11: 61.2% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. lower-+.f64 61.5

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

5. Applied rewrites 61.5%

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

Alternative 12: 20.8% accurate, 212.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. lower-+.f64 61.5

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

5. Applied rewrites 61.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 27.3%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024308 
(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))))