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

Percentage Accurate: 99.9% → 99.9%

Time: 11.5s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

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: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ t_1 := z \cdot \sin y\\ t_2 := t\_0 - t\_1\\ t_3 := x - t\_1\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+20}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y)))
        (t_1 (* z (sin y)))
        (t_2 (- t_0 t_1))
        (t_3 (- x t_1)))
   (if (<= t_2 -2e+20) t_3 (if (<= t_2 2e+15) t_0 t_3))))

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 t_3 = x - t_1;
	double tmp;
	if (t_2 <= -2e+20) {
		tmp = t_3;
	} else if (t_2 <= 2e+15) {
		tmp = t_0;
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_0 = x + cos(y)
    t_1 = z * sin(y)
    t_2 = t_0 - t_1
    t_3 = x - t_1
    if (t_2 <= (-2d+20)) then
        tmp = t_3
    else if (t_2 <= 2d+15) then
        tmp = t_0
    else
        tmp = t_3
    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 t_3 = x - t_1;
	double tmp;
	if (t_2 <= -2e+20) {
		tmp = t_3;
	} else if (t_2 <= 2e+15) {
		tmp = t_0;
	} else {
		tmp = t_3;
	}
	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
	t_3 = x - t_1
	tmp = 0
	if t_2 <= -2e+20:
		tmp = t_3
	elif t_2 <= 2e+15:
		tmp = t_0
	else:
		tmp = t_3
	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)
	t_3 = Float64(x - t_1)
	tmp = 0.0
	if (t_2 <= -2e+20)
		tmp = t_3;
	elseif (t_2 <= 2e+15)
		tmp = t_0;
	else
		tmp = t_3;
	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;
	t_3 = x - t_1;
	tmp = 0.0;
	if (t_2 <= -2e+20)
		tmp = t_3;
	elseif (t_2 <= 2e+15)
		tmp = t_0;
	else
		tmp = t_3;
	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]}, Block[{t$95$3 = N[(x - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+20], t$95$3, If[LessEqual[t$95$2, 2e+15], t$95$0, t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -2e20 or 2e15 < (-.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 x around inf

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

   5. Simplified 99.9%

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

   if -2e20 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 2e15

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

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

      3. cos-lowering-cos.f64 99.1

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

   5. Simplified 99.1%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -2 \cdot 10^{+20}:\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{elif}\;\left(x + \cos y\right) - z \cdot \sin y \leq 2 \cdot 10^{+15}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \cos y\right) - z \cdot \sin y\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 0.99:\\ \;\;\;\;\cos y\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(y, z, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (cos y)) (* z (sin y)))))
   (if (<= t_0 -2.0) x (if (<= t_0 0.99) (cos y) (- x (fma y z -1.0))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + cos(y)) - (z * sin(y));
	double tmp;
	if (t_0 <= -2.0) {
		tmp = x;
	} else if (t_0 <= 0.99) {
		tmp = cos(y);
	} else {
		tmp = x - fma(y, z, -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + cos(y)) - Float64(z * sin(y)))
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = x;
	elseif (t_0 <= 0.99)
		tmp = cos(y);
	else
		tmp = Float64(x - fma(y, z, -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], x, If[LessEqual[t$95$0, 0.99], N[Cos[y], $MachinePrecision], N[(x - N[(y * z + -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 59.3%

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

   if -2 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.98999999999999999

   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. --lowering--.f64 N/A

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

      2. cos-lowering-cos.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sin-lowering-sin.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around 0

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

      1. cos-lowering-cos.f64 98.9

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

   8. Simplified 98.9%

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

   if 0.98999999999999999 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. --lowering--.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 74.9

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

   5. Simplified 74.9%

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

Alternative 4: 82.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{+136}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+164}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) (- z))))
   (if (<= z -1.75e+136) t_0 (if (<= z 3.2e+164) (+ x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * -z;
	double tmp;
	if (z <= -1.75e+136) {
		tmp = t_0;
	} else if (z <= 3.2e+164) {
		tmp = x + cos(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 = sin(y) * -z
    if (z <= (-1.75d+136)) then
        tmp = t_0
    else if (z <= 3.2d+164) then
        tmp = x + cos(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.sin(y) * -z;
	double tmp;
	if (z <= -1.75e+136) {
		tmp = t_0;
	} else if (z <= 3.2e+164) {
		tmp = x + Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) * -z
	tmp = 0
	if z <= -1.75e+136:
		tmp = t_0
	elif z <= 3.2e+164:
		tmp = x + math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * Float64(-z))
	tmp = 0.0
	if (z <= -1.75e+136)
		tmp = t_0;
	elseif (z <= 3.2e+164)
		tmp = Float64(x + cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) * -z;
	tmp = 0.0;
	if (z <= -1.75e+136)
		tmp = t_0;
	elseif (z <= 3.2e+164)
		tmp = x + cos(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision]}, If[LessEqual[z, -1.75e+136], t$95$0, If[LessEqual[z, 3.2e+164], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.75000000000000001e136 or 3.1999999999999998e164 < 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sin-lowering-sin.f64 73.7

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

   5. Simplified 73.7%

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

   if -1.75000000000000001e136 < z < 3.1999999999999998e164

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

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

      3. cos-lowering-cos.f64 91.3

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

   5. Simplified 91.3%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+136}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+164}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ \mathbf{if}\;y \leq -740:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.052:\\ \;\;\;\;1 + \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot 0.16666666666666666, -0.5\right), -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y))))
   (if (<= y -740.0)
     t_0
     (if (<= y 0.052)
       (+ 1.0 (fma y (fma y (fma y (* z 0.16666666666666666) -0.5) (- z)) x))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + cos(y);
	double tmp;
	if (y <= -740.0) {
		tmp = t_0;
	} else if (y <= 0.052) {
		tmp = 1.0 + fma(y, fma(y, fma(y, (z * 0.16666666666666666), -0.5), -z), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + cos(y))
	tmp = 0.0
	if (y <= -740.0)
		tmp = t_0;
	elseif (y <= 0.052)
		tmp = Float64(1.0 + fma(y, fma(y, fma(y, Float64(z * 0.16666666666666666), -0.5), Float64(-z)), x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -740.0], t$95$0, If[LessEqual[y, 0.052], N[(1.0 + N[(y * N[(y * N[(y * N[(z * 0.16666666666666666), $MachinePrecision] + -0.5), $MachinePrecision] + (-z)), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -740 or 0.0519999999999999976 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

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

      3. cos-lowering-cos.f64 66.0

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

   5. Simplified 66.0%

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

   if -740 < y < 0.0519999999999999976

   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. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. neg-lowering-neg.f64 98.7

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

   5. Simplified 98.7%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -740:\\ \;\;\;\;x + \cos y\\ \mathbf{elif}\;y \leq 0.052:\\ \;\;\;\;1 + \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot 0.16666666666666666, -0.5\right), -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \frac{1}{z} \cdot \frac{z}{x}, x\right)\\ \mathbf{if}\;y \leq -63000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.6:\\ \;\;\;\;1 + \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot 0.16666666666666666, -0.5\right), -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma x (* (/ 1.0 z) (/ z x)) x)))
   (if (<= y -63000000.0)
     t_0
     (if (<= y 3.6)
       (+ 1.0 (fma y (fma y (fma y (* z 0.16666666666666666) -0.5) (- z)) x))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(x, ((1.0 / z) * (z / x)), x);
	double tmp;
	if (y <= -63000000.0) {
		tmp = t_0;
	} else if (y <= 3.6) {
		tmp = 1.0 + fma(y, fma(y, fma(y, (z * 0.16666666666666666), -0.5), -z), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(x, Float64(Float64(1.0 / z) * Float64(z / x)), x)
	tmp = 0.0
	if (y <= -63000000.0)
		tmp = t_0;
	elseif (y <= 3.6)
		tmp = Float64(1.0 + fma(y, fma(y, fma(y, Float64(z * 0.16666666666666666), -0.5), Float64(-z)), x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(N[(1.0 / z), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -63000000.0], t$95$0, If[LessEqual[y, 3.6], N[(1.0 + N[(y * N[(y * N[(y * N[(z * 0.16666666666666666), $MachinePrecision] + -0.5), $MachinePrecision] + (-z)), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.3e7 or 3.60000000000000009 < y

   1. Initial program 99.8%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. clear-num N/A

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

      5. flip-- N/A

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

      6. /-lowering-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate--l+ N/A

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

      9. +-lowering-+.f64 N/A

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

      10. cos-lowering-cos.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. sin-lowering-sin.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. distribute-lft-out-- N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. cos-lowering-cos.f64 N/A

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

      11. sin-lowering-sin.f64 87.5

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

   7. Simplified 87.5%

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

   8. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. cos-lowering-cos.f64 N/A

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

      11. sin-lowering-sin.f64 N/A

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

      12. /-lowering-/.f64 89.6

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

   10. Simplified 89.6%

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

   11. Taylor expanded in y around 0

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{z}} \cdot \frac{z}{x}, x\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 44.6

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

   13. Simplified 44.6%

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

   if -6.3e7 < y < 3.60000000000000009

   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. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. neg-lowering-neg.f64 97.9

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

   5. Simplified 97.9%

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

Alternative 7: 70.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -29500:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 3.6:\\ \;\;\;\;1 + \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot 0.16666666666666666, -0.5\right), -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -29500.0)
   (+ x 1.0)
   (if (<= y 3.6)
     (+ 1.0 (fma y (fma y (fma y (* z 0.16666666666666666) -0.5) (- z)) x))
     (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -29500.0) {
		tmp = x + 1.0;
	} else if (y <= 3.6) {
		tmp = 1.0 + fma(y, fma(y, fma(y, (z * 0.16666666666666666), -0.5), -z), x);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -29500.0)
		tmp = Float64(x + 1.0);
	elseif (y <= 3.6)
		tmp = Float64(1.0 + fma(y, fma(y, fma(y, Float64(z * 0.16666666666666666), -0.5), Float64(-z)), x));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -29500.0], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 3.6], N[(1.0 + N[(y * N[(y * N[(y * N[(z * 0.16666666666666666), $MachinePrecision] + -0.5), $MachinePrecision] + (-z)), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -29500 or 3.60000000000000009 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 44.3

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

   5. Simplified 44.3%

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

   if -29500 < y < 3.60000000000000009

   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. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. neg-lowering-neg.f64 98.7

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

   5. Simplified 98.7%

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

Alternative 8: 70.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8000000000:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+25}:\\ \;\;\;\;x - \mathsf{fma}\left(y, z, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8000000000.0)
   (+ x 1.0)
   (if (<= y 1.25e+25) (- x (fma y z -1.0)) (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8000000000.0) {
		tmp = x + 1.0;
	} else if (y <= 1.25e+25) {
		tmp = x - fma(y, z, -1.0);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8000000000.0)
		tmp = Float64(x + 1.0);
	elseif (y <= 1.25e+25)
		tmp = Float64(x - fma(y, z, -1.0));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8000000000.0], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 1.25e+25], N[(x - N[(y * z + -1.0), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -8e9 or 1.25000000000000006e25 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 45.0

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

   5. Simplified 45.0%

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

   if -8e9 < y < 1.25000000000000006e25

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. --lowering--.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 95.9

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

   5. Simplified 95.9%

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

Alternative 9: 65.5% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0068:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-210}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.0068)
   (+ x 1.0)
   (if (<= x 1.8e-210) (fma y (- z) 1.0) (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.0068) {
		tmp = x + 1.0;
	} else if (x <= 1.8e-210) {
		tmp = fma(y, -z, 1.0);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.0068)
		tmp = Float64(x + 1.0);
	elseif (x <= 1.8e-210)
		tmp = fma(y, Float64(-z), 1.0);
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.0068], N[(x + 1.0), $MachinePrecision], If[LessEqual[x, 1.8e-210], N[(y * (-z) + 1.0), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00679999999999999962 or 1.7999999999999999e-210 < 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. +-commutative N/A

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

      2. +-lowering-+.f64 74.2

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

   5. Simplified 74.2%

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

   if -0.00679999999999999962 < x < 1.7999999999999999e-210

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   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. --lowering--.f64 N/A

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

      2. cos-lowering-cos.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sin-lowering-sin.f64 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-lowering-neg.f64 53.0

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

   8. Simplified 53.0%

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

Alternative 10: 61.3% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.8:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 820:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= x -0.8) x (if (<= x 820.0) 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.8) {
		tmp = x;
	} else if (x <= 820.0) {
		tmp = 1.0;
	} else {
		tmp = 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 <= (-0.8d0)) then
        tmp = x
    else if (x <= 820.0d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.8) {
		tmp = x;
	} else if (x <= 820.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -0.8:
		tmp = x
	elif x <= 820.0:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.8)
		tmp = x;
	elseif (x <= 820.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -0.8)
		tmp = x;
	elseif (x <= 820.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.8], x, If[LessEqual[x, 820.0], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if x < -0.80000000000000004 or 820 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 78.2%

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

   if -0.80000000000000004 < x < 820

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 43.9

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

   5. Simplified 43.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 43.2%

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

Alternative 11: 62.0% accurate, 53.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + 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 = x + 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x + 1.0), $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. +-commutative N/A

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

   2. +-lowering-+.f64 62.6

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

5. Simplified 62.6%

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

Alternative 12: 22.0% 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. +-commutative N/A

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

   2. +-lowering-+.f64 62.6

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

5. Simplified 62.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 21.8%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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))))