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

Percentage Accurate: 99.9% → 99.9%

Time: 11.3s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + sin(y)) + (z * cos(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 + sin(y)) + (z * cos(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + sin(y)) + (z * cos(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 + sin(y)) + (z * cos(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 80.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \sin y\right) + \cos y \cdot z\\ \mathbf{if}\;t\_0 \leq -400000:\\ \;\;\;\;z + x\\ \mathbf{elif}\;t\_0 \leq -0.005:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;t\_0 \leq 10^{-27}:\\ \;\;\;\;y + \left(z + x\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (+ x (sin y)) (* (cos y) z))))
   (if (<= t_0 -400000.0)
     (+ z x)
     (if (<= t_0 -0.005)
       (sin y)
       (if (<= t_0 1e-27) (+ y (+ z x)) (if (<= t_0 1.0) (sin y) (+ z x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + sin(y)) + (cos(y) * z);
	double tmp;
	if (t_0 <= -400000.0) {
		tmp = z + x;
	} else if (t_0 <= -0.005) {
		tmp = sin(y);
	} else if (t_0 <= 1e-27) {
		tmp = y + (z + x);
	} else if (t_0 <= 1.0) {
		tmp = sin(y);
	} else {
		tmp = z + 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) :: t_0
    real(8) :: tmp
    t_0 = (x + sin(y)) + (cos(y) * z)
    if (t_0 <= (-400000.0d0)) then
        tmp = z + x
    else if (t_0 <= (-0.005d0)) then
        tmp = sin(y)
    else if (t_0 <= 1d-27) then
        tmp = y + (z + x)
    else if (t_0 <= 1.0d0) then
        tmp = sin(y)
    else
        tmp = z + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + Math.sin(y)) + (Math.cos(y) * z);
	double tmp;
	if (t_0 <= -400000.0) {
		tmp = z + x;
	} else if (t_0 <= -0.005) {
		tmp = Math.sin(y);
	} else if (t_0 <= 1e-27) {
		tmp = y + (z + x);
	} else if (t_0 <= 1.0) {
		tmp = Math.sin(y);
	} else {
		tmp = z + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + math.sin(y)) + (math.cos(y) * z)
	tmp = 0
	if t_0 <= -400000.0:
		tmp = z + x
	elif t_0 <= -0.005:
		tmp = math.sin(y)
	elif t_0 <= 1e-27:
		tmp = y + (z + x)
	elif t_0 <= 1.0:
		tmp = math.sin(y)
	else:
		tmp = z + x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + sin(y)) + Float64(cos(y) * z))
	tmp = 0.0
	if (t_0 <= -400000.0)
		tmp = Float64(z + x);
	elseif (t_0 <= -0.005)
		tmp = sin(y);
	elseif (t_0 <= 1e-27)
		tmp = Float64(y + Float64(z + x));
	elseif (t_0 <= 1.0)
		tmp = sin(y);
	else
		tmp = Float64(z + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + sin(y)) + (cos(y) * z);
	tmp = 0.0;
	if (t_0 <= -400000.0)
		tmp = z + x;
	elseif (t_0 <= -0.005)
		tmp = sin(y);
	elseif (t_0 <= 1e-27)
		tmp = y + (z + x);
	elseif (t_0 <= 1.0)
		tmp = sin(y);
	else
		tmp = z + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -400000.0], N[(z + x), $MachinePrecision], If[LessEqual[t$95$0, -0.005], N[Sin[y], $MachinePrecision], If[LessEqual[t$95$0, 1e-27], N[(y + N[(z + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[y], $MachinePrecision], N[(z + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -4e5 or 1 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 72.3

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

   5. Simplified 72.3%

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

   if -4e5 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.0050000000000000001 or 1e-27 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 95.6

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

   5. Simplified 95.6%

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

   6. Taylor expanded in x around 0

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

      1. lower-sin.f64 89.8

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

   8. Simplified 89.8%

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

   if -0.0050000000000000001 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 1e-27

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \sin y\right) + \cos y \cdot z \leq -400000:\\ \;\;\;\;z + x\\ \mathbf{elif}\;\left(x + \sin y\right) + \cos y \cdot z \leq -0.005:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;\left(x + \sin y\right) + \cos y \cdot z \leq 10^{-27}:\\ \;\;\;\;y + \left(z + x\right)\\ \mathbf{elif}\;\left(x + \sin y\right) + \cos y \cdot z \leq 1:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + \sin y\right) + \cos y \cdot z \leq -4 \cdot 10^{+40}:\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;y + \left(z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ (+ x (sin y)) (* (cos y) z)) -4e+40) (+ z x) (+ y (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x + sin(y)) + (cos(y) * z)) <= -4e+40) {
		tmp = z + x;
	} else {
		tmp = y + (z + 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 (((x + sin(y)) + (cos(y) * z)) <= (-4d+40)) then
        tmp = z + x
    else
        tmp = y + (z + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x + Math.sin(y)) + (Math.cos(y) * z)) <= -4e+40) {
		tmp = z + x;
	} else {
		tmp = y + (z + x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x + math.sin(y)) + (math.cos(y) * z)) <= -4e+40:
		tmp = z + x
	else:
		tmp = y + (z + x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(x + sin(y)) + Float64(cos(y) * z)) <= -4e+40)
		tmp = Float64(z + x);
	else
		tmp = Float64(y + Float64(z + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x + sin(y)) + (cos(y) * z)) <= -4e+40)
		tmp = z + x;
	else
		tmp = y + (z + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], -4e+40], N[(z + x), $MachinePrecision], N[(y + N[(z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -4.00000000000000012e40

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 74.8

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

   5. Simplified 74.8%

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

   if -4.00000000000000012e40 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 60.0

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

   5. Simplified 60.0%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \sin y\right) + \cos y \cdot z \leq -4 \cdot 10^{+40}:\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;y + \left(z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(\cos y + \frac{x}{z}\right)\\ \mathbf{if}\;z \leq -6.1 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.0085:\\ \;\;\;\;z + \left(x + \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (+ (cos y) (/ x z)))))
   (if (<= z -6.1e+26) t_0 (if (<= z 0.0085) (+ z (+ x (sin y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (cos(y) + (x / z));
	double tmp;
	if (z <= -6.1e+26) {
		tmp = t_0;
	} else if (z <= 0.0085) {
		tmp = z + (x + sin(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * (cos(y) + (x / z))
    if (z <= (-6.1d+26)) then
        tmp = t_0
    else if (z <= 0.0085d0) then
        tmp = z + (x + sin(y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (Math.cos(y) + (x / z));
	double tmp;
	if (z <= -6.1e+26) {
		tmp = t_0;
	} else if (z <= 0.0085) {
		tmp = z + (x + Math.sin(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (math.cos(y) + (x / z))
	tmp = 0
	if z <= -6.1e+26:
		tmp = t_0
	elif z <= 0.0085:
		tmp = z + (x + math.sin(y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(cos(y) + Float64(x / z)))
	tmp = 0.0
	if (z <= -6.1e+26)
		tmp = t_0;
	elseif (z <= 0.0085)
		tmp = Float64(z + Float64(x + sin(y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (cos(y) + (x / z));
	tmp = 0.0;
	if (z <= -6.1e+26)
		tmp = t_0;
	elseif (z <= 0.0085)
		tmp = z + (x + sin(y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[Cos[y], $MachinePrecision] + N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.1e+26], t$95$0, If[LessEqual[z, 0.0085], N[(z + N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.1000000000000003e26 or 0.0085000000000000006 < z

   1. Initial program 99.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. distribute-lft-out N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sin.f64 99.8

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 99.7

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

   10. Simplified 99.7%

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

   if -6.1000000000000003e26 < z < 0.0085000000000000006

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.1 \cdot 10^{+26}:\\ \;\;\;\;z \cdot \left(\cos y + \frac{x}{z}\right)\\ \mathbf{elif}\;z \leq 0.0085:\\ \;\;\;\;z + \left(x + \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\cos y + \frac{x}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{+178}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 31000000000000:\\ \;\;\;\;z + \left(x + \sin y\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+99}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)))
   (if (<= z -6.8e+178)
     t_0
     (if (<= z 31000000000000.0)
       (+ z (+ x (sin y)))
       (if (<= z 2e+99) t_0 (fma (cos y) z (+ y x)))))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double tmp;
	if (z <= -6.8e+178) {
		tmp = t_0;
	} else if (z <= 31000000000000.0) {
		tmp = z + (x + sin(y));
	} else if (z <= 2e+99) {
		tmp = t_0;
	} else {
		tmp = fma(cos(y), z, (y + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * z)
	tmp = 0.0
	if (z <= -6.8e+178)
		tmp = t_0;
	elseif (z <= 31000000000000.0)
		tmp = Float64(z + Float64(x + sin(y)));
	elseif (z <= 2e+99)
		tmp = t_0;
	else
		tmp = fma(cos(y), z, Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -6.8e+178], t$95$0, If[LessEqual[z, 31000000000000.0], N[(z + N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e+99], t$95$0, N[(N[Cos[y], $MachinePrecision] * z + N[(y + x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.8000000000000005e178 or 3.1e13 < z < 1.9999999999999999e99

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 92.2

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

   5. Simplified 92.2%

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

   if -6.8000000000000005e178 < z < 3.1e13

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 93.4%

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

   if 1.9999999999999999e99 < z

   1. Initial program 99.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 88.7

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

   7. Simplified 88.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+178}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;z \leq 31000000000000:\\ \;\;\;\;z + \left(x + \sin y\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{+178}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.14 \cdot 10^{-48}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;z \leq 0.36:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)))
   (if (<= z -6.8e+178)
     t_0
     (if (<= z -1.14e-48) (+ z x) (if (<= z 0.36) (+ x (sin y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double tmp;
	if (z <= -6.8e+178) {
		tmp = t_0;
	} else if (z <= -1.14e-48) {
		tmp = z + x;
	} else if (z <= 0.36) {
		tmp = x + sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(y) * z
    if (z <= (-6.8d+178)) then
        tmp = t_0
    else if (z <= (-1.14d-48)) then
        tmp = z + x
    else if (z <= 0.36d0) then
        tmp = x + sin(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cos(y) * z;
	double tmp;
	if (z <= -6.8e+178) {
		tmp = t_0;
	} else if (z <= -1.14e-48) {
		tmp = z + x;
	} else if (z <= 0.36) {
		tmp = x + Math.sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cos(y) * z
	tmp = 0
	if z <= -6.8e+178:
		tmp = t_0
	elif z <= -1.14e-48:
		tmp = z + x
	elif z <= 0.36:
		tmp = x + math.sin(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * z)
	tmp = 0.0
	if (z <= -6.8e+178)
		tmp = t_0;
	elseif (z <= -1.14e-48)
		tmp = Float64(z + x);
	elseif (z <= 0.36)
		tmp = Float64(x + sin(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cos(y) * z;
	tmp = 0.0;
	if (z <= -6.8e+178)
		tmp = t_0;
	elseif (z <= -1.14e-48)
		tmp = z + x;
	elseif (z <= 0.36)
		tmp = x + sin(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -6.8e+178], t$95$0, If[LessEqual[z, -1.14e-48], N[(z + x), $MachinePrecision], If[LessEqual[z, 0.36], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.8000000000000005e178 or 0.35999999999999999 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 82.3

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

   5. Simplified 82.3%

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

   if -6.8000000000000005e178 < z < -1.1400000000000001e-48

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 76.7

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

   5. Simplified 76.7%

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

   if -1.1400000000000001e-48 < z < 0.35999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 94.0

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

   5. Simplified 94.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+178}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;z \leq -1.14 \cdot 10^{-48}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;z \leq 0.36:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos y, z, y + x\right)\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{-48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (cos y) z (+ y x))))
   (if (<= z -1.05e-48) t_0 (if (<= z 4.2e-6) (+ x (sin y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(cos(y), z, (y + x));
	double tmp;
	if (z <= -1.05e-48) {
		tmp = t_0;
	} else if (z <= 4.2e-6) {
		tmp = x + sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(cos(y), z, Float64(y + x))
	tmp = 0.0
	if (z <= -1.05e-48)
		tmp = t_0;
	elseif (z <= 4.2e-6)
		tmp = Float64(x + sin(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.05e-48], t$95$0, If[LessEqual[z, 4.2e-6], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.04999999999999994e-48 or 4.1999999999999996e-6 < z

   1. Initial program 99.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 80.6

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

   7. Simplified 80.6%

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

   if -1.04999999999999994e-48 < z < 4.1999999999999996e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 93.9

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

   5. Simplified 93.9%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, y + x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \sin y\\ \mathbf{if}\;y \leq -0.054:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1000000:\\ \;\;\;\;z + \mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot -0.5, 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (sin y))))
   (if (<= y -0.054)
     t_0
     (if (<= y 1000000.0) (+ z (fma y (fma y (* z -0.5) 1.0) x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + sin(y);
	double tmp;
	if (y <= -0.054) {
		tmp = t_0;
	} else if (y <= 1000000.0) {
		tmp = z + fma(y, fma(y, (z * -0.5), 1.0), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + sin(y))
	tmp = 0.0
	if (y <= -0.054)
		tmp = t_0;
	elseif (y <= 1000000.0)
		tmp = Float64(z + fma(y, fma(y, Float64(z * -0.5), 1.0), x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.054], t$95$0, If[LessEqual[y, 1000000.0], N[(z + N[(y * N[(y * N[(z * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0539999999999999994 or 1e6 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 54.7

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

   5. Simplified 54.7%

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

   if -0.0539999999999999994 < y < 1e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 98.5

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

   5. Simplified 98.5%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.054:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;y \leq 1000000:\\ \;\;\;\;z + \mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot -0.5, 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+22}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 1.76 \cdot 10^{+31}:\\ \;\;\;\;z + \mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot -0.5, 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.8e+22)
   (+ z x)
   (if (<= y 1.76e+31) (+ z (fma y (fma y (* z -0.5) 1.0) x)) (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.8e+22) {
		tmp = z + x;
	} else if (y <= 1.76e+31) {
		tmp = z + fma(y, fma(y, (z * -0.5), 1.0), x);
	} else {
		tmp = z + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.8e+22)
		tmp = Float64(z + x);
	elseif (y <= 1.76e+31)
		tmp = Float64(z + fma(y, fma(y, Float64(z * -0.5), 1.0), x));
	else
		tmp = Float64(z + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.8e+22], N[(z + x), $MachinePrecision], If[LessEqual[y, 1.76e+31], N[(z + N[(y * N[(y * N[(z * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.8e22 or 1.76e31 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 31.3

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

   5. Simplified 31.3%

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

   if -5.8e22 < y < 1.76e31

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.4

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

   5. Simplified 91.4%

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

Alternative 10: 66.4% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 62.6

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

5. Simplified 62.6%

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

Alternative 11: 37.9% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-sin.f64 54.9

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

5. Simplified 54.9%

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

6. Taylor expanded in y around 0

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

   1. lower-+.f64 37.6

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

8. Simplified 37.6%

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

9. Final simplification 37.6%

   \[\leadsto y + x \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, C"
  :precision binary64
  (+ (+ x (sin y)) (* z (cos y))))