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

Percentage Accurate: 99.9% → 99.9%

Time: 8.5s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot z\\ t_1 := \left(\cos y + x\right) - t\_0\\ t_2 := \left(1 + x\right) - t\_0\\ \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.998:\\ \;\;\;\;\cos y - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) z))
        (t_1 (- (+ (cos y) x) t_0))
        (t_2 (- (+ 1.0 x) t_0)))
   (if (<= t_1 -1000.0) t_2 (if (<= t_1 0.998) (- (cos y) t_0) t_2))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * z;
	double t_1 = (cos(y) + x) - t_0;
	double t_2 = (1.0 + x) - t_0;
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.998) {
		tmp = cos(y) - t_0;
	} else {
		tmp = t_2;
	}
	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 = sin(y) * z
    t_1 = (cos(y) + x) - t_0
    t_2 = (1.0d0 + x) - t_0
    if (t_1 <= (-1000.0d0)) then
        tmp = t_2
    else if (t_1 <= 0.998d0) then
        tmp = cos(y) - t_0
    else
        tmp = t_2
    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 = (Math.cos(y) + x) - t_0;
	double t_2 = (1.0 + x) - t_0;
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.998) {
		tmp = Math.cos(y) - t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) * z
	t_1 = (math.cos(y) + x) - t_0
	t_2 = (1.0 + x) - t_0
	tmp = 0
	if t_1 <= -1000.0:
		tmp = t_2
	elif t_1 <= 0.998:
		tmp = math.cos(y) - t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * z)
	t_1 = Float64(Float64(cos(y) + x) - t_0)
	t_2 = Float64(Float64(1.0 + x) - t_0)
	tmp = 0.0
	if (t_1 <= -1000.0)
		tmp = t_2;
	elseif (t_1 <= 0.998)
		tmp = Float64(cos(y) - t_0);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) * z;
	t_1 = (cos(y) + x) - t_0;
	t_2 = (1.0 + x) - t_0;
	tmp = 0.0;
	if (t_1 <= -1000.0)
		tmp = t_2;
	elseif (t_1 <= 0.998)
		tmp = cos(y) - t_0;
	else
		tmp = t_2;
	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[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 + x), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -1000.0], t$95$2, If[LessEqual[t$95$1, 0.998], N[(N[Cos[y], $MachinePrecision] - t$95$0), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -1e3 or 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 99.8%

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

   if -1e3 < (-.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 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos y + x\right) - \sin y \cdot z \leq -1000:\\ \;\;\;\;\left(1 + x\right) - \sin y \cdot z\\ \mathbf{elif}\;\left(\cos y + x\right) - \sin y \cdot z \leq 0.998:\\ \;\;\;\;\cos y - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - \sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y + x\\ t_1 := t\_0 - \sin y \cdot z\\ t_2 := \mathsf{fma}\left(-\sin y, z, x\right)\\ \mathbf{if}\;t\_1 \leq -2000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (cos y) x))
        (t_1 (- t_0 (* (sin y) z)))
        (t_2 (fma (- (sin y)) z x)))
   (if (<= t_1 -2000000000.0) t_2 (if (<= t_1 2.0) t_0 t_2))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) + x;
	double t_1 = t_0 - (sin(y) * z);
	double t_2 = fma(-sin(y), z, x);
	double tmp;
	if (t_1 <= -2000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) + x)
	t_1 = Float64(t_0 - Float64(sin(y) * z))
	t_2 = fma(Float64(-sin(y)), z, x)
	tmp = 0.0
	if (t_1 <= -2000000000.0)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sin[y], $MachinePrecision]) * z + x), $MachinePrecision]}, If[LessEqual[t$95$1, -2000000000.0], t$95$2, If[LessEqual[t$95$1, 2.0], t$95$0, t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -2e9 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. Taylor expanded in y around 0

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 99.5%

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

   if -2e9 < (-.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 z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 97.8

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

   5. Applied rewrites 97.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos y + x\right) - \sin y \cdot z \leq -2000000000:\\ \;\;\;\;\mathsf{fma}\left(-\sin y, z, x\right)\\ \mathbf{elif}\;\left(\cos y + x\right) - \sin y \cdot z \leq 2:\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\sin y, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos y + x\right) - \sin y \cdot z\\ t_1 := x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.998:\\ \;\;\;\;\cos y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (cos y) x) (* (sin y) z))) (t_1 (- x (fma z y -1.0))))
   (if (<= t_0 -1000.0) t_1 (if (<= t_0 0.998) (cos y) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (cos(y) + x) - (sin(y) * z);
	double t_1 = x - fma(z, y, -1.0);
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.998) {
		tmp = cos(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(cos(y) + x) - Float64(sin(y) * z))
	t_1 = Float64(x - fma(z, y, -1.0))
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = t_1;
	elseif (t_0 <= 0.998)
		tmp = cos(y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1000.0], t$95$1, If[LessEqual[t$95$0, 0.998], N[Cos[y], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -1e3 or 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

      \[\leadsto \color{blue}{1 + \left(x + -1 \cdot \left(y \cdot z\right)\right)} \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   if -1e3 < (-.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 z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.9%

      \[\leadsto \cos y \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos y + x\right) - \sin y \cdot z \leq -1000:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{elif}\;\left(\cos y + x\right) - \sin y \cdot z \leq 0.998:\\ \;\;\;\;\cos y\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + x\right) - \sin y \cdot z\\ \mathbf{if}\;z \leq -55000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ 1.0 x) (* (sin y) z))))
   (if (<= z -55000000000000.0) t_0 (if (<= z 1.8e-25) (+ (cos y) 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 <= -55000000000000.0) {
		tmp = t_0;
	} else if (z <= 1.8e-25) {
		tmp = cos(y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + x) - (sin(y) * z)
    if (z <= (-55000000000000.0d0)) then
        tmp = t_0
    else if (z <= 1.8d-25) then
        tmp = cos(y) + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (1.0 + x) - (Math.sin(y) * z);
	double tmp;
	if (z <= -55000000000000.0) {
		tmp = t_0;
	} else if (z <= 1.8e-25) {
		tmp = Math.cos(y) + 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 <= -55000000000000.0:
		tmp = t_0
	elif z <= 1.8e-25:
		tmp = math.cos(y) + 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 <= -55000000000000.0)
		tmp = t_0;
	elseif (z <= 1.8e-25)
		tmp = Float64(cos(y) + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (1.0 + x) - (sin(y) * z);
	tmp = 0.0;
	if (z <= -55000000000000.0)
		tmp = t_0;
	elseif (z <= 1.8e-25)
		tmp = cos(y) + 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, -55000000000000.0], t$95$0, If[LessEqual[z, 1.8e-25], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5e13 or 1.8e-25 < 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

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

   5. Applied rewrites 99.3%

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

   if -5.5e13 < z < 1.8e-25

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -55000000000000:\\ \;\;\;\;\left(1 + x\right) - \sin y \cdot z\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - \sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot \sin y\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{+135}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+92}:\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) (sin y))))
   (if (<= z -7.5e+135) t_0 (if (<= z 8.8e+92) (+ (cos y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -z * sin(y);
	double tmp;
	if (z <= -7.5e+135) {
		tmp = t_0;
	} else if (z <= 8.8e+92) {
		tmp = cos(y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -z * sin(y)
    if (z <= (-7.5d+135)) then
        tmp = t_0
    else if (z <= 8.8d+92) then
        tmp = cos(y) + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -z * Math.sin(y);
	double tmp;
	if (z <= -7.5e+135) {
		tmp = t_0;
	} else if (z <= 8.8e+92) {
		tmp = Math.cos(y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z * math.sin(y)
	tmp = 0
	if z <= -7.5e+135:
		tmp = t_0
	elif z <= 8.8e+92:
		tmp = math.cos(y) + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) * sin(y))
	tmp = 0.0
	if (z <= -7.5e+135)
		tmp = t_0;
	elseif (z <= 8.8e+92)
		tmp = Float64(cos(y) + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z * sin(y);
	tmp = 0.0;
	if (z <= -7.5e+135)
		tmp = t_0;
	elseif (z <= 8.8e+92)
		tmp = cos(y) + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.5e+135], t$95$0, If[LessEqual[z, 8.8e+92], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.49999999999999947e135 or 8.79999999999999969e92 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. 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 71.6

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

   5. Applied rewrites 71.6%

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

   if -7.49999999999999947e135 < z < 8.79999999999999969e92

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 87.8

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

   5. Applied rewrites 87.8%

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

Alternative 7: 81.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y + x\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.195:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right) \cdot z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (cos y) x)))
   (if (<= y -1.7e+19)
     t_0
     (if (<= y 0.195)
       (fma (* (fma 0.16666666666666666 (* y y) -1.0) z) y (+ 1.0 x))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) + x;
	double tmp;
	if (y <= -1.7e+19) {
		tmp = t_0;
	} else if (y <= 0.195) {
		tmp = fma((fma(0.16666666666666666, (y * y), -1.0) * z), y, (1.0 + x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) + x)
	tmp = 0.0
	if (y <= -1.7e+19)
		tmp = t_0;
	elseif (y <= 0.195)
		tmp = fma(Float64(fma(0.16666666666666666, Float64(y * y), -1.0) * 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[Cos[y], $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.7e+19], t$95$0, If[LessEqual[y, 0.195], N[(N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + -1.0), $MachinePrecision] * z), $MachinePrecision] * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7e19 or 0.19500000000000001 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 61.6

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

   5. Applied rewrites 61.6%

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

   if -1.7e19 < y < 0.19500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(x + y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right)\right)} \]
   10. 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. sub-neg N/A

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

       3. mul-1-neg N/A

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

       4. +-commutative N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. sub-neg N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. mul-1-neg N/A

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

       17. lower-neg.f64 N/A

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

       18. lower-+.f64 99.2

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

   11. Applied rewrites 99.2%

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

   12. Taylor expanded in z around inf

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

   14. Applied rewrites 99.2%

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

Alternative 8: 70.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+24}:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5\right) \cdot y - z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6.5e+24)
   (+ 1.0 x)
   (if (<= y 5.5e+27)
     (fma (- (* (fma 0.16666666666666666 (* z y) -0.5) y) z) y (+ 1.0 x))
     (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.5e+24) {
		tmp = 1.0 + x;
	} else if (y <= 5.5e+27) {
		tmp = fma(((fma(0.16666666666666666, (z * y), -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 <= -6.5e+24)
		tmp = Float64(1.0 + x);
	elseif (y <= 5.5e+27)
		tmp = fma(Float64(Float64(fma(0.16666666666666666, Float64(z * y), -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, -6.5e+24], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 5.5e+27], N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $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 < -6.4999999999999996e24 or 5.49999999999999966e27 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 40.2

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

   5. Applied rewrites 40.2%

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

   if -6.4999999999999996e24 < y < 5.49999999999999966e27

   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 95.4

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

   5. Applied rewrites 95.4%

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

Alternative 9: 70.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+24}:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right) \cdot z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.2e+24)
   (+ 1.0 x)
   (if (<= y 4.3e+16)
     (fma (* (fma 0.16666666666666666 (* y y) -1.0) z) y (+ 1.0 x))
     (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.2e+24) {
		tmp = 1.0 + x;
	} else if (y <= 4.3e+16) {
		tmp = fma((fma(0.16666666666666666, (y * y), -1.0) * z), y, (1.0 + x));
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.2e+24)
		tmp = Float64(1.0 + x);
	elseif (y <= 4.3e+16)
		tmp = fma(Float64(fma(0.16666666666666666, Float64(y * y), -1.0) * z), y, Float64(1.0 + x));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.2e+24], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 4.3e+16], N[(N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + -1.0), $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 < -3.1999999999999997e24 or 4.3e16 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 39.7

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

   5. Applied rewrites 39.7%

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

   if -3.1999999999999997e24 < y < 4.3e16

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(x + y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right)\right)} \]
   10. 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. sub-neg N/A

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

       3. mul-1-neg N/A

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

       4. +-commutative N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. sub-neg N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. mul-1-neg N/A

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

       17. lower-neg.f64 N/A

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

       18. lower-+.f64 96.9

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

   11. Applied rewrites 96.9%

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

   12. Taylor expanded in z around inf

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

   14. Applied rewrites 96.8%

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

Alternative 10: 69.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+152}:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+22}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.1e+152)
   (+ 1.0 x)
   (if (<= y 4e+22) (- x (fma z y -1.0)) (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.1e+152) {
		tmp = 1.0 + x;
	} else if (y <= 4e+22) {
		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 <= -1.1e+152)
		tmp = Float64(1.0 + x);
	elseif (y <= 4e+22)
		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, -1.1e+152], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 4e+22], N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.0999999999999999e152 or 4e22 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 39.5

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

   5. Applied rewrites 39.5%

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

   if -1.0999999999999999e152 < y < 4e22

   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 88.7

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

   5. Applied rewrites 88.7%

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

Alternative 11: 62.5% 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 56.6

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

5. Applied rewrites 56.6%

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

Alternative 12: 22.2% 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 56.6

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

5. Applied rewrites 56.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 19.5%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(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))))