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

Percentage Accurate: 99.9% → 99.9%

Time: 8.1s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 92.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ t_1 := z \cdot \sin y\\ t_2 := t\_0 - t\_1\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+17} \lor \neg \left(t\_2 \leq 1.001\right):\\ \;\;\;\;\left(x + 1\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y))) (t_1 (* z (sin y))) (t_2 (- t_0 t_1)))
   (if (or (<= t_2 -1e+17) (not (<= t_2 1.001)))
     (- (+ x 1.0) t_1)
     (- t_0 (* z y)))))

C

double code(double x, double y, double z) {
	double t_0 = x + cos(y);
	double t_1 = z * sin(y);
	double t_2 = t_0 - t_1;
	double tmp;
	if ((t_2 <= -1e+17) || !(t_2 <= 1.001)) {
		tmp = (x + 1.0) - t_1;
	} else {
		tmp = t_0 - (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x + cos(y)
    t_1 = z * sin(y)
    t_2 = t_0 - t_1
    if ((t_2 <= (-1d+17)) .or. (.not. (t_2 <= 1.001d0))) then
        tmp = (x + 1.0d0) - t_1
    else
        tmp = t_0 - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + Math.cos(y);
	double t_1 = z * Math.sin(y);
	double t_2 = t_0 - t_1;
	double tmp;
	if ((t_2 <= -1e+17) || !(t_2 <= 1.001)) {
		tmp = (x + 1.0) - t_1;
	} else {
		tmp = t_0 - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + math.cos(y)
	t_1 = z * math.sin(y)
	t_2 = t_0 - t_1
	tmp = 0
	if (t_2 <= -1e+17) or not (t_2 <= 1.001):
		tmp = (x + 1.0) - t_1
	else:
		tmp = t_0 - (z * y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + cos(y))
	t_1 = Float64(z * sin(y))
	t_2 = Float64(t_0 - t_1)
	tmp = 0.0
	if ((t_2 <= -1e+17) || !(t_2 <= 1.001))
		tmp = Float64(Float64(x + 1.0) - t_1);
	else
		tmp = Float64(t_0 - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + cos(y);
	t_1 = z * sin(y);
	t_2 = t_0 - t_1;
	tmp = 0.0;
	if ((t_2 <= -1e+17) || ~((t_2 <= 1.001)))
		tmp = (x + 1.0) - t_1;
	else
		tmp = t_0 - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - t$95$1), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -1e+17], N[Not[LessEqual[t$95$2, 1.001]], $MachinePrecision]], N[(N[(x + 1.0), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 - N[(z * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -1e17 or 1.0009999999999999 < (-.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 99.7%

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

   if -1e17 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 1.0009999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.8

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

   5. Applied rewrites 77.8%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -1 \cdot 10^{+17} \lor \neg \left(\left(x + \cos y\right) - z \cdot \sin y \leq 1.001\right):\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\left(x + \cos y\right) - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := \left(x + \cos y\right) - t\_0\\ \mathbf{if}\;t\_1 \leq -4000000000 \lor \neg \left(t\_1 \leq 1.001\right):\\ \;\;\;\;\left(x + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos y - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (- (+ x (cos y)) t_0)))
   (if (or (<= t_1 -4000000000.0) (not (<= t_1 1.001)))
     (- (+ x 1.0) t_0)
     (- (cos y) (* z y)))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double t_1 = (x + cos(y)) - t_0;
	double tmp;
	if ((t_1 <= -4000000000.0) || !(t_1 <= 1.001)) {
		tmp = (x + 1.0) - t_0;
	} else {
		tmp = cos(y) - (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * sin(y)
    t_1 = (x + cos(y)) - t_0
    if ((t_1 <= (-4000000000.0d0)) .or. (.not. (t_1 <= 1.001d0))) then
        tmp = (x + 1.0d0) - t_0
    else
        tmp = cos(y) - (z * y)
    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 t_1 = (x + Math.cos(y)) - t_0;
	double tmp;
	if ((t_1 <= -4000000000.0) || !(t_1 <= 1.001)) {
		tmp = (x + 1.0) - t_0;
	} else {
		tmp = Math.cos(y) - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.sin(y)
	t_1 = (x + math.cos(y)) - t_0
	tmp = 0
	if (t_1 <= -4000000000.0) or not (t_1 <= 1.001):
		tmp = (x + 1.0) - t_0
	else:
		tmp = math.cos(y) - (z * y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	t_1 = Float64(Float64(x + cos(y)) - t_0)
	tmp = 0.0
	if ((t_1 <= -4000000000.0) || !(t_1 <= 1.001))
		tmp = Float64(Float64(x + 1.0) - t_0);
	else
		tmp = Float64(cos(y) - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * sin(y);
	t_1 = (x + cos(y)) - t_0;
	tmp = 0.0;
	if ((t_1 <= -4000000000.0) || ~((t_1 <= 1.001)))
		tmp = (x + 1.0) - t_0;
	else
		tmp = cos(y) - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -4000000000.0], N[Not[LessEqual[t$95$1, 1.001]], $MachinePrecision]], N[(N[(x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -4e9 or 1.0009999999999999 < (-.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 99.7%

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

   if -4e9 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 1.0009999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.9

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-cos.f64 75.1

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

   8. Applied rewrites 75.1%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -4000000000 \lor \neg \left(\left(x + \cos y\right) - z \cdot \sin y \leq 1.001\right):\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{-6} \lor \neg \left(x \leq 1.25 \cdot 10^{-21}\right):\\ \;\;\;\;\left(x + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos y - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))))
   (if (or (<= x -1.3e-6) (not (<= x 1.25e-21)))
     (- (+ x 1.0) t_0)
     (- (cos y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double tmp;
	if ((x <= -1.3e-6) || !(x <= 1.25e-21)) {
		tmp = (x + 1.0) - t_0;
	} else {
		tmp = cos(y) - 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 ((x <= (-1.3d-6)) .or. (.not. (x <= 1.25d-21))) then
        tmp = (x + 1.0d0) - t_0
    else
        tmp = cos(y) - 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 ((x <= -1.3e-6) || !(x <= 1.25e-21)) {
		tmp = (x + 1.0) - t_0;
	} else {
		tmp = Math.cos(y) - t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.sin(y)
	tmp = 0
	if (x <= -1.3e-6) or not (x <= 1.25e-21):
		tmp = (x + 1.0) - t_0
	else:
		tmp = math.cos(y) - t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	tmp = 0.0
	if ((x <= -1.3e-6) || !(x <= 1.25e-21))
		tmp = Float64(Float64(x + 1.0) - t_0);
	else
		tmp = Float64(cos(y) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * sin(y);
	tmp = 0.0;
	if ((x <= -1.3e-6) || ~((x <= 1.25e-21)))
		tmp = (x + 1.0) - t_0;
	else
		tmp = cos(y) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -1.3e-6], N[Not[LessEqual[x, 1.25e-21]], $MachinePrecision]], N[(N[(x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.30000000000000005e-6 or 1.24999999999999993e-21 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.7%

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

   if -1.30000000000000005e-6 < x < 1.24999999999999993e-21

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-cos.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-6} \lor \neg \left(x \leq 1.25 \cdot 10^{-21}\right):\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-6} \lor \neg \left(x \leq 5.6 \cdot 10^{-6}\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\cos y - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.4e-6) (not (<= x 5.6e-6))) (+ 1.0 x) (- (cos y) (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4e-6) || !(x <= 5.6e-6)) {
		tmp = 1.0 + x;
	} else {
		tmp = cos(y) - (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.4d-6)) .or. (.not. (x <= 5.6d-6))) then
        tmp = 1.0d0 + x
    else
        tmp = cos(y) - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4e-6) || !(x <= 5.6e-6)) {
		tmp = 1.0 + x;
	} else {
		tmp = Math.cos(y) - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.4e-6) or not (x <= 5.6e-6):
		tmp = 1.0 + x
	else:
		tmp = math.cos(y) - (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.4e-6) || !(x <= 5.6e-6))
		tmp = Float64(1.0 + x);
	else
		tmp = Float64(cos(y) - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.4e-6) || ~((x <= 5.6e-6)))
		tmp = 1.0 + x;
	else
		tmp = cos(y) - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.4e-6], N[Not[LessEqual[x, 5.6e-6]], $MachinePrecision]], N[(1.0 + x), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.39999999999999994e-6 or 5.59999999999999975e-6 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 86.5

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

   5. Applied rewrites 86.5%

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

   if -1.39999999999999994e-6 < x < 5.59999999999999975e-6

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-cos.f64 61.0

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

   8. Applied rewrites 61.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-6} \lor \neg \left(x \leq 5.6 \cdot 10^{-6}\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\cos y - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+137} \lor \neg \left(z \leq 5.1 \cdot 10^{+147}\right):\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.3e+137) (not (<= z 5.1e+147))) (* (- z) (sin y)) (+ 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.3e+137) || !(z <= 5.1e+147)) {
		tmp = -z * sin(y);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.3d+137)) .or. (.not. (z <= 5.1d+147))) then
        tmp = -z * sin(y)
    else
        tmp = 1.0d0 + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.3e+137) || !(z <= 5.1e+147)) {
		tmp = -z * Math.sin(y);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.3e+137) or not (z <= 5.1e+147):
		tmp = -z * math.sin(y)
	else:
		tmp = 1.0 + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.3e+137) || !(z <= 5.1e+147))
		tmp = Float64(Float64(-z) * sin(y));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.3e+137) || ~((z <= 5.1e+147)))
		tmp = -z * sin(y);
	else
		tmp = 1.0 + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.3e+137], N[Not[LessEqual[z, 5.1e+147]], $MachinePrecision]], N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e137 or 5.09999999999999999e147 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 71.2

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

   5. Applied rewrites 71.2%

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

   if -1.3e137 < z < 5.09999999999999999e147

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 74.6

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

   5. Applied rewrites 74.6%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+137} \lor \neg \left(z \leq 5.1 \cdot 10^{+147}\right):\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -36000000000 \lor \neg \left(y \leq 15.5\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + \mathsf{fma}\left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5, y \cdot y, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -36000000000.0) (not (<= y 15.5)))
   (+ 1.0 x)
   (-
    (+ x (fma (- (* 0.041666666666666664 (* y y)) 0.5) (* y y) 1.0))
    (*
     (fma
      (* z (fma 0.008333333333333333 (* y y) -0.16666666666666666))
      (* y y)
      z)
     y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -36000000000.0) || !(y <= 15.5)) {
		tmp = 1.0 + x;
	} else {
		tmp = (x + fma(((0.041666666666666664 * (y * y)) - 0.5), (y * y), 1.0)) - (fma((z * fma(0.008333333333333333, (y * y), -0.16666666666666666)), (y * y), z) * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -36000000000.0) || !(y <= 15.5))
		tmp = Float64(1.0 + x);
	else
		tmp = Float64(Float64(x + fma(Float64(Float64(0.041666666666666664 * Float64(y * y)) - 0.5), Float64(y * y), 1.0)) - Float64(fma(Float64(z * fma(0.008333333333333333, Float64(y * y), -0.16666666666666666)), Float64(y * y), z) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -36000000000.0], N[Not[LessEqual[y, 15.5]], $MachinePrecision]], N[(1.0 + x), $MachinePrecision], N[(N[(x + N[(N[(N[(0.041666666666666664 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(z * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.6e10 or 15.5 < 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 44.8

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

   5. Applied rewrites 44.8%

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

   if -3.6e10 < y < 15.5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(x + \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)\right)}\right) - z \cdot y \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right) + 1\right)}\right) - z \cdot y \]

      2. *-commutative N/A

         \[\leadsto \left(x + \left(\color{blue}{\left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right) \cdot {y}^{2}} + 1\right)\right) - z \cdot y \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(x + \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)}\right) - z \cdot y \]

      4. lower--.f64 N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}}, {y}^{2}, 1\right)\right) - z \cdot y \]

      5. lower-*.f64 N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {y}^{2}} - \frac{1}{2}, {y}^{2}, 1\right)\right) - z \cdot y \]

      6. unpow2 N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot \color{blue}{\left(y \cdot y\right)} - \frac{1}{2}, {y}^{2}, 1\right)\right) - z \cdot y \]

      7. lower-*.f64 N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot \color{blue}{\left(y \cdot y\right)} - \frac{1}{2}, {y}^{2}, 1\right)\right) - z \cdot y \]

      8. unpow2 N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}, \color{blue}{y \cdot y}, 1\right)\right) - z \cdot y \]

      9. lower-*.f64 98.5

         \[\leadsto \left(x + \mathsf{fma}\left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5, \color{blue}{y \cdot y}, 1\right)\right) - z \cdot y \]

   8. Applied rewrites 98.5%

      \[\leadsto \left(x + \color{blue}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5, y \cdot y, 1\right)}\right) - z \cdot y \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}, y \cdot y, 1\right)\right) - \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}, y \cdot y, 1\right)\right) - \color{blue}{\left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right) \cdot y} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}, y \cdot y, 1\right)\right) - \color{blue}{\left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right) \cdot y} \]

       3. +-commutative N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}, y \cdot y, 1\right)\right) - \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right) + z\right)} \cdot y \]

       4. *-commutative N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}, y \cdot y, 1\right)\right) - \left(\color{blue}{\left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right) \cdot {y}^{2}} + z\right) \cdot y \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}, y \cdot y, 1\right)\right) - \color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right), {y}^{2}, z\right)} \cdot y \]

       6. +-commutative N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}, y \cdot y, 1\right)\right) - \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot \left({y}^{2} \cdot z\right) + \frac{-1}{6} \cdot z}, {y}^{2}, z\right) \cdot y \]

       7. associate-*r* N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}, y \cdot y, 1\right)\right) - \mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot z} + \frac{-1}{6} \cdot z, {y}^{2}, z\right) \cdot y \]

       8. distribute-rgt-out N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}, y \cdot y, 1\right)\right) - \mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right)}, {y}^{2}, z\right) \cdot y \]

       9. lower-*.f64 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}, y \cdot y, 1\right)\right) - \mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right)}, {y}^{2}, z\right) \cdot y \]

       10. lower-fma.f64 N/A

           \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}, y \cdot y, 1\right)\right) - \mathsf{fma}\left(z \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {y}^{2}, \frac{-1}{6}\right)}, {y}^{2}, z\right) \cdot y \]

       11. unpow2 N/A

           \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}, y \cdot y, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{y \cdot y}, \frac{-1}{6}\right), {y}^{2}, z\right) \cdot y \]

       12. lower-*.f64 N/A

           \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}, y \cdot y, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{y \cdot y}, \frac{-1}{6}\right), {y}^{2}, z\right) \cdot y \]

       13. unpow2 N/A

           \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}, y \cdot y, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \color{blue}{y \cdot y}, z\right) \cdot y \]

       14. lower-*.f64 99.0

           \[\leadsto \left(x + \mathsf{fma}\left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5, y \cdot y, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), \color{blue}{y \cdot y}, z\right) \cdot y \]

   11. Applied rewrites 99.0%

       \[\leadsto \left(x + \mathsf{fma}\left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5, y \cdot y, 1\right)\right) - \color{blue}{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -36000000000 \lor \neg \left(y \leq 15.5\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + \mathsf{fma}\left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5, y \cdot y, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+16} \lor \neg \left(y \leq 750\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\right) - \mathsf{fma}\left(z, y, \left(\left(\left(y \cdot y\right) \cdot z\right) \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right)\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.65e+16) (not (<= y 750.0)))
   (+ 1.0 x)
   (-
    (+ x (fma (* y y) -0.5 1.0))
    (fma
     z
     y
     (*
      (* (* (* y y) z) (fma (* y y) 0.008333333333333333 -0.16666666666666666))
      y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.65e+16) || !(y <= 750.0)) {
		tmp = 1.0 + x;
	} else {
		tmp = (x + fma((y * y), -0.5, 1.0)) - fma(z, y, ((((y * y) * z) * fma((y * y), 0.008333333333333333, -0.16666666666666666)) * y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.65e+16) || !(y <= 750.0))
		tmp = Float64(1.0 + x);
	else
		tmp = Float64(Float64(x + fma(Float64(y * y), -0.5, 1.0)) - fma(z, y, Float64(Float64(Float64(Float64(y * y) * z) * fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666)) * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.65e+16], N[Not[LessEqual[y, 750.0]], $MachinePrecision]], N[(1.0 + x), $MachinePrecision], N[(N[(x + N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] - N[(z * y + N[(N[(N[(N[(y * y), $MachinePrecision] * z), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.65e16 or 750 < 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 44.7

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

   5. Applied rewrites 44.7%

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

   if -1.65e16 < y < 750

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 97.9

          \[\leadsto \left(x + \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right) \cdot z, \color{blue}{y \cdot y}, z\right) \cdot y \]

   8. Applied rewrites 97.9%

      \[\leadsto \left(x + \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\right) - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right) \cdot z, y \cdot y, z\right) \cdot y} \]
   9. Step-by-step derivation

   10. Applied rewrites 97.9%

       \[\leadsto \left(x + \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\right) - \mathsf{fma}\left(z, \color{blue}{y}, \left(\left(\left(y \cdot y\right) \cdot z\right) \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right)\right) \cdot y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+16} \lor \neg \left(y \leq 750\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\right) - \mathsf{fma}\left(z, y, \left(\left(\left(y \cdot y\right) \cdot z\right) \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right)\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+16} \lor \neg \left(y \leq 750\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right) \cdot z, y \cdot y, z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.65e+16) (not (<= y 750.0)))
   (+ 1.0 x)
   (-
    (+ x (fma (* y y) -0.5 1.0))
    (*
     (fma
      (* (fma 0.008333333333333333 (* y y) -0.16666666666666666) z)
      (* y y)
      z)
     y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.65e+16) || !(y <= 750.0)) {
		tmp = 1.0 + x;
	} else {
		tmp = (x + fma((y * y), -0.5, 1.0)) - (fma((fma(0.008333333333333333, (y * y), -0.16666666666666666) * z), (y * y), z) * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.65e+16) || !(y <= 750.0))
		tmp = Float64(1.0 + x);
	else
		tmp = Float64(Float64(x + fma(Float64(y * y), -0.5, 1.0)) - Float64(fma(Float64(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666) * z), Float64(y * y), z) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.65e+16], N[Not[LessEqual[y, 750.0]], $MachinePrecision]], N[(1.0 + x), $MachinePrecision], N[(N[(x + N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * z), $MachinePrecision] * N[(y * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.65e16 or 750 < 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 44.7

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

   5. Applied rewrites 44.7%

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

   if -1.65e16 < y < 750

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 97.9

          \[\leadsto \left(x + \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right) \cdot z, \color{blue}{y \cdot y}, z\right) \cdot y \]

   8. Applied rewrites 97.9%

      \[\leadsto \left(x + \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\right) - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right) \cdot z, y \cdot y, z\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+16} \lor \neg \left(y \leq 750\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right) \cdot z, y \cdot y, z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+28} \lor \neg \left(y \leq 8500\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot z\right) \cdot y - 0.5, y, -z\right), y, 1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.25e+28) (not (<= y 8500.0)))
   (+ 1.0 x)
   (fma (fma (- (* (* 0.16666666666666666 z) y) 0.5) y (- z)) y (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.25e+28) || !(y <= 8500.0)) {
		tmp = 1.0 + x;
	} else {
		tmp = fma(fma((((0.16666666666666666 * z) * y) - 0.5), y, -z), y, (1.0 + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.25e+28) || !(y <= 8500.0))
		tmp = Float64(1.0 + x);
	else
		tmp = fma(fma(Float64(Float64(Float64(0.16666666666666666 * z) * y) - 0.5), y, Float64(-z)), y, Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.25e+28], N[Not[LessEqual[y, 8500.0]], $MachinePrecision]], N[(1.0 + x), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * z), $MachinePrecision] * y), $MachinePrecision] - 0.5), $MachinePrecision] * y + (-z)), $MachinePrecision] * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.24999999999999989e28 or 8500 < 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 44.7

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

   5. Applied rewrites 44.7%

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

   if -1.24999999999999989e28 < y < 8500

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 97.2%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+28} \lor \neg \left(y \leq 8500\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot z\right) \cdot y - 0.5, y, -z\right), y, 1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+16} \lor \neg \left(y \leq 750\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - z, y, 1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.65e+16) (not (<= y 750.0)))
   (+ 1.0 x)
   (fma (- (* -0.5 y) z) y (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.65e+16) || !(y <= 750.0)) {
		tmp = 1.0 + x;
	} else {
		tmp = fma(((-0.5 * y) - z), y, (1.0 + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.65e+16) || !(y <= 750.0))
		tmp = Float64(1.0 + x);
	else
		tmp = fma(Float64(Float64(-0.5 * y) - z), y, Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.65e+16], N[Not[LessEqual[y, 750.0]], $MachinePrecision]], N[(1.0 + x), $MachinePrecision], N[(N[(N[(-0.5 * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.65e16 or 750 < 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 44.7

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

   5. Applied rewrites 44.7%

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

   if -1.65e16 < y < 750

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+16} \lor \neg \left(y \leq 750\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - z, y, 1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.1% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.25 \cdot 10^{+16} \lor \neg \left(y \leq 15.5\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.25e+16) (not (<= y 15.5)))
   (+ 1.0 x)
   (fma (- z) y (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.25e+16) || !(y <= 15.5)) {
		tmp = 1.0 + x;
	} else {
		tmp = fma(-z, y, (1.0 + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.25e+16) || !(y <= 15.5))
		tmp = Float64(1.0 + x);
	else
		tmp = fma(Float64(-z), y, Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.25e+16], N[Not[LessEqual[y, 15.5]], $MachinePrecision]], N[(1.0 + x), $MachinePrecision], N[((-z) * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.25e16 or 15.5 < 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 44.7

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

   5. Applied rewrites 44.7%

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

   if -3.25e16 < y < 15.5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-+.f64 97.2

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

   5. Applied rewrites 97.2%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.25 \cdot 10^{+16} \lor \neg \left(y \leq 15.5\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.1% 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 63.1

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

5. Applied rewrites 63.1%

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

Alternative 14: 21.9% accurate, 212.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. lower-+.f64 63.1

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

5. Applied rewrites 63.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 19.7%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024329 
(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))))