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

Percentage Accurate: 99.9% → 99.9%

Time: 8.5s

Alternatives: 14

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.9

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 92.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ t_1 := t\_0 - z \cdot \sin y\\ \mathbf{if}\;t\_1 \leq -10000000 \lor \neg \left(t\_1 \leq 0.995\right):\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y))) (t_1 (- t_0 (* z (sin y)))))
   (if (or (<= t_1 -10000000.0) (not (<= t_1 0.995)))
     (fma (sin y) (- z) (+ 1.0 x))
     (- t_0 (* z y)))))

C

double code(double x, double y, double z) {
	double t_0 = x + cos(y);
	double t_1 = t_0 - (z * sin(y));
	double tmp;
	if ((t_1 <= -10000000.0) || !(t_1 <= 0.995)) {
		tmp = fma(sin(y), -z, (1.0 + x));
	} else {
		tmp = t_0 - (z * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + cos(y))
	t_1 = Float64(t_0 - Float64(z * sin(y)))
	tmp = 0.0
	if ((t_1 <= -10000000.0) || !(t_1 <= 0.995))
		tmp = fma(sin(y), Float64(-z), Float64(1.0 + x));
	else
		tmp = Float64(t_0 - Float64(z * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -10000000.0], N[Not[LessEqual[t$95$1, 0.995]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] * (-z) + N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(z * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -1e7 or 0.994999999999999996 < (-.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 98.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 98.5

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 98.5

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

   7. Applied rewrites 98.5%

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

   if -1e7 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.994999999999999996

   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 54.0

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

   5. Applied rewrites 54.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -10000000 \lor \neg \left(\left(x + \cos y\right) - z \cdot \sin y \leq 0.995\right):\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \cos y\right) - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ t_1 := z \cdot \sin y\\ t_2 := t\_0 - t\_1\\ \mathbf{if}\;t\_2 \leq -10000000 \lor \neg \left(t\_2 \leq 0.995\right):\\ \;\;\;\;\left(x + 1\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y))) (t_1 (* z (sin y))) (t_2 (- t_0 t_1)))
   (if (or (<= t_2 -10000000.0) (not (<= t_2 0.995)))
     (- (+ x 1.0) t_1)
     (- t_0 (* z y)))))

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 tmp;
	if ((t_2 <= -10000000.0) || !(t_2 <= 0.995)) {
		tmp = (x + 1.0) - t_1;
	} else {
		tmp = t_0 - (z * y);
	}
	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) :: tmp
    t_0 = x + cos(y)
    t_1 = z * sin(y)
    t_2 = t_0 - t_1
    if ((t_2 <= (-10000000.0d0)) .or. (.not. (t_2 <= 0.995d0))) then
        tmp = (x + 1.0d0) - t_1
    else
        tmp = t_0 - (z * y)
    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 tmp;
	if ((t_2 <= -10000000.0) || !(t_2 <= 0.995)) {
		tmp = (x + 1.0) - t_1;
	} else {
		tmp = t_0 - (z * y);
	}
	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
	tmp = 0
	if (t_2 <= -10000000.0) or not (t_2 <= 0.995):
		tmp = (x + 1.0) - t_1
	else:
		tmp = t_0 - (z * y)
	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)
	tmp = 0.0
	if ((t_2 <= -10000000.0) || !(t_2 <= 0.995))
		tmp = Float64(Float64(x + 1.0) - t_1);
	else
		tmp = Float64(t_0 - Float64(z * y));
	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;
	tmp = 0.0;
	if ((t_2 <= -10000000.0) || ~((t_2 <= 0.995)))
		tmp = (x + 1.0) - t_1;
	else
		tmp = t_0 - (z * y);
	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]}, If[Or[LessEqual[t$95$2, -10000000.0], N[Not[LessEqual[t$95$2, 0.995]], $MachinePrecision]], N[(N[(x + 1.0), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 - N[(z * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -1e7 or 0.994999999999999996 < (-.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 98.5%

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

   if -1e7 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.994999999999999996

   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 54.0

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

   5. Applied rewrites 54.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -10000000 \lor \neg \left(\left(x + \cos y\right) - z \cdot \sin y \leq 0.995\right):\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\left(x + \cos y\right) - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-12}:\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(-z, \sin y, \cos y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.9e-12)
   (- (+ x 1.0) (* z (sin y)))
   (if (<= x 4.6e-18)
     (fma (- z) (sin y) (cos y))
     (fma (sin y) (- z) (+ 1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.9e-12) {
		tmp = (x + 1.0) - (z * sin(y));
	} else if (x <= 4.6e-18) {
		tmp = fma(-z, sin(y), cos(y));
	} else {
		tmp = fma(sin(y), -z, (1.0 + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.9e-12)
		tmp = Float64(Float64(x + 1.0) - Float64(z * sin(y)));
	elseif (x <= 4.6e-18)
		tmp = fma(Float64(-z), sin(y), cos(y));
	else
		tmp = fma(sin(y), Float64(-z), Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.9e-12], N[(N[(x + 1.0), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.6e-18], N[((-z) * N[Sin[y], $MachinePrecision] + N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * (-z) + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.89999999999999998e-12

   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 96.9%

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

   if -1.89999999999999998e-12 < x < 4.6000000000000002e-18

   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. fp-cancel-sub-sign-inv N/A

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-cos.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 4.6000000000000002e-18 < 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

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

   5. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 99.8

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.8

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

   7. Applied rewrites 99.8%

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

Alternative 5: 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 6: 69.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+242}:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq -9.4 \cdot 10^{+86}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{elif}\;y \leq 155000000:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.85e+242)
   (+ 1.0 x)
   (if (<= y -9.4e+86)
     (* (- z) (sin y))
     (if (<= y 155000000.0) (fma (- z) y (+ 1.0 x)) (+ 1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.85e+242) {
		tmp = 1.0 + x;
	} else if (y <= -9.4e+86) {
		tmp = -z * sin(y);
	} else if (y <= 155000000.0) {
		tmp = fma(-z, y, (1.0 + x));
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.85e+242)
		tmp = Float64(1.0 + x);
	elseif (y <= -9.4e+86)
		tmp = Float64(Float64(-z) * sin(y));
	elseif (y <= 155000000.0)
		tmp = fma(Float64(-z), y, Float64(1.0 + x));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.85e+242], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, -9.4e+86], N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 155000000.0], N[((-z) * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.85e242 or 1.55e8 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 60.2

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

   5. Applied rewrites 60.2%

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

   if -1.85e242 < y < -9.4000000000000004e86

   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 57.8

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

   5. Applied rewrites 57.8%

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

   if -9.4000000000000004e86 < y < 1.55e8

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-+.f64 88.3

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

   5. Applied rewrites 88.3%

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

Alternative 7: 88.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + 1.0) - (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 + 1.0d0) - (z * sin(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x + 1.0), $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. 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 89.5%

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

Alternative 8: 69.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+15} \lor \neg \left(y \leq 8.2 \cdot 10^{-30}\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot z\right) \cdot y - 0.5, y, -z\right), y, 1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.3e+15) (not (<= y 8.2e-30)))
   (+ 1.0 x)
   (fma (fma (- (* (* 0.16666666666666666 z) y) 0.5) y (- z)) y (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.3e+15) || !(y <= 8.2e-30)) {
		tmp = 1.0 + x;
	} else {
		tmp = fma(fma((((0.16666666666666666 * z) * y) - 0.5), y, -z), y, (1.0 + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.3e+15) || !(y <= 8.2e-30))
		tmp = Float64(1.0 + x);
	else
		tmp = fma(fma(Float64(Float64(Float64(0.16666666666666666 * z) * y) - 0.5), y, Float64(-z)), y, Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.3e+15], N[Not[LessEqual[y, 8.2e-30]], $MachinePrecision]], N[(1.0 + x), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * z), $MachinePrecision] * y), $MachinePrecision] - 0.5), $MachinePrecision] * y + (-z)), $MachinePrecision] * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e15 or 8.2000000000000007e-30 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 49.5

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

   5. Applied rewrites 49.5%

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

   if -1.3e15 < y < 8.2000000000000007e-30

   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. Applied rewrites 96.3%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+15} \lor \neg \left(y \leq 8.2 \cdot 10^{-30}\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot z\right) \cdot y - 0.5, y, -z\right), y, 1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+15} \lor \neg \left(y \leq 8.2 \cdot 10^{-30}\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - z, y, 1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.35e+15) (not (<= y 8.2e-30)))
   (+ 1.0 x)
   (fma (- (* -0.5 y) z) y (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.35e+15) || !(y <= 8.2e-30)) {
		tmp = 1.0 + x;
	} else {
		tmp = fma(((-0.5 * y) - z), y, (1.0 + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.35e+15) || !(y <= 8.2e-30))
		tmp = Float64(1.0 + x);
	else
		tmp = fma(Float64(Float64(-0.5 * y) - z), y, Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.35e+15], N[Not[LessEqual[y, 8.2e-30]], $MachinePrecision]], N[(1.0 + x), $MachinePrecision], N[(N[(N[(-0.5 * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35e15 or 8.2000000000000007e-30 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 49.5

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

   5. Applied rewrites 49.5%

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

   if -1.35e15 < y < 8.2000000000000007e-30

   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 96.2

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

   5. Applied rewrites 96.2%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+15} \lor \neg \left(y \leq 8.2 \cdot 10^{-30}\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - z, y, 1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.3% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+24} \lor \neg \left(y \leq 155000000\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4e+24) (not (<= y 155000000.0)))
   (+ 1.0 x)
   (fma (- z) y (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4e+24) || !(y <= 155000000.0)) {
		tmp = 1.0 + x;
	} else {
		tmp = fma(-z, y, (1.0 + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4e+24) || !(y <= 155000000.0))
		tmp = Float64(1.0 + x);
	else
		tmp = fma(Float64(-z), y, Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4e+24], N[Not[LessEqual[y, 155000000.0]], $MachinePrecision]], N[(1.0 + x), $MachinePrecision], N[((-z) * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.9999999999999999e24 or 1.55e8 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 48.2

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

   5. Applied rewrites 48.2%

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

   if -3.9999999999999999e24 < y < 1.55e8

   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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-+.f64 93.2

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

   5. Applied rewrites 93.2%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+24} \lor \neg \left(y \leq 155000000\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.3% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.116 \lor \neg \left(x \leq 11000\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -0.116) (not (<= x 11000.0))) (+ 1.0 x) (fma (- z) y 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.116) || !(x <= 11000.0)) {
		tmp = 1.0 + x;
	} else {
		tmp = fma(-z, y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -0.116) || !(x <= 11000.0))
		tmp = Float64(1.0 + x);
	else
		tmp = fma(Float64(-z), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -0.116], N[Not[LessEqual[x, 11000.0]], $MachinePrecision]], N[(1.0 + x), $MachinePrecision], N[((-z) * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.116000000000000006 or 11000 < 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 85.9

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

   5. Applied rewrites 85.9%

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

   if -0.116000000000000006 < x < 11000

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 55.2

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.8%

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

   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 55.8%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.116 \lor \neg \left(x \leq 11000\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.0% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+223}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.2e+223) (* (- y) z) (+ 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.2e+223) {
		tmp = -y * z;
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.2d+223)) then
        tmp = -y * z
    else
        tmp = 1.0d0 + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.2e+223) {
		tmp = -y * z;
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.2e+223:
		tmp = -y * z
	else:
		tmp = 1.0 + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.2e+223)
		tmp = Float64(Float64(-y) * z);
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.2e+223)
		tmp = -y * z;
	else
		tmp = 1.0 + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.2e+223], N[((-y) * z), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2e223

   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 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 40.9%

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

   if -2.2e223 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 65.3

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

   5. Applied rewrites 65.3%

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

Alternative 13: 62.6% 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 60.7

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

5. Applied rewrites 60.7%

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

Alternative 14: 22.3% accurate, 212.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. lower-+.f64 60.7

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

5. Applied rewrites 60.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 20.7%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024339 
(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))))