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

Percentage Accurate: 99.9% → 99.9%

Time: 12.8s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

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

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

   10. lower-fma.f64 N/A

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

   11. lower-neg.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ t_1 := \sin y \cdot z\\ t_2 := t\_0 - t\_1\\ t_3 := x - t\_1\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+23}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 100000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y)))
        (t_1 (* (sin y) z))
        (t_2 (- t_0 t_1))
        (t_3 (- x t_1)))
   (if (<= t_2 -1e+23) t_3 (if (<= t_2 100000000.0) t_0 t_3))))

C

double code(double x, double y, double z) {
	double t_0 = x + cos(y);
	double t_1 = sin(y) * z;
	double t_2 = t_0 - t_1;
	double t_3 = x - t_1;
	double tmp;
	if (t_2 <= -1e+23) {
		tmp = t_3;
	} else if (t_2 <= 100000000.0) {
		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 = sin(y) * z
    t_2 = t_0 - t_1
    t_3 = x - t_1
    if (t_2 <= (-1d+23)) then
        tmp = t_3
    else if (t_2 <= 100000000.0d0) 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 = Math.sin(y) * z;
	double t_2 = t_0 - t_1;
	double t_3 = x - t_1;
	double tmp;
	if (t_2 <= -1e+23) {
		tmp = t_3;
	} else if (t_2 <= 100000000.0) {
		tmp = t_0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + math.cos(y)
	t_1 = math.sin(y) * z
	t_2 = t_0 - t_1
	t_3 = x - t_1
	tmp = 0
	if t_2 <= -1e+23:
		tmp = t_3
	elif t_2 <= 100000000.0:
		tmp = t_0
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + cos(y))
	t_1 = Float64(sin(y) * z)
	t_2 = Float64(t_0 - t_1)
	t_3 = Float64(x - t_1)
	tmp = 0.0
	if (t_2 <= -1e+23)
		tmp = t_3;
	elseif (t_2 <= 100000000.0)
		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 = sin(y) * z;
	t_2 = t_0 - t_1;
	t_3 = x - t_1;
	tmp = 0.0;
	if (t_2 <= -1e+23)
		tmp = t_3;
	elseif (t_2 <= 100000000.0)
		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[(N[Sin[y], $MachinePrecision] * z), $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, -1e+23], t$95$3, If[LessEqual[t$95$2, 100000000.0], 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))) < -9.9999999999999992e22 or 1e8 < (-.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

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.8

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

   5. Simplified 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-sin.f64 N/A

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

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

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 99.8

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. sub-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. /-rgt-identity N/A

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

      13. metadata-eval N/A

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

      14. rgt-mult-inverse N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. distribute-neg-frac2 N/A

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

      17. mul-1-neg N/A

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

      18. associate-*r/ N/A

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

      19. *-rgt-identity N/A

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

      20. times-frac N/A

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

      21. *-lft-identity N/A

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

   10. Simplified 99.8%

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

   11. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 99.8

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

   13. Simplified 99.8%

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

   if -9.9999999999999992e22 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 1e8

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 99.2

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

   5. Simplified 99.2%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - \sin y \cdot z \leq -1 \cdot 10^{+23}:\\ \;\;\;\;x - \sin y \cdot z\\ \mathbf{elif}\;\left(x + \cos y\right) - \sin y \cdot z \leq 100000000:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;x - \sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \cos y\right) - \sin y \cdot z\\ t_1 := x - \mathsf{fma}\left(y, z, -1\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.9999790455489881:\\ \;\;\;\;\cos y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (cos y)) (* (sin y) z))) (t_1 (- x (fma y z -1.0))))
   (if (<= t_0 -1e+27) t_1 (if (<= t_0 0.9999790455489881) (cos y) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (x + cos(y)) - (sin(y) * z);
	double t_1 = x - fma(y, z, -1.0);
	double tmp;
	if (t_0 <= -1e+27) {
		tmp = t_1;
	} else if (t_0 <= 0.9999790455489881) {
		tmp = cos(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + cos(y)) - Float64(sin(y) * z))
	t_1 = Float64(x - fma(y, z, -1.0))
	tmp = 0.0
	if (t_0 <= -1e+27)
		tmp = t_1;
	elseif (t_0 <= 0.9999790455489881)
		tmp = cos(y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x - N[(y * z + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+27], t$95$1, If[LessEqual[t$95$0, 0.9999790455489881], N[Cos[y], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -1e27 or 0.99997904554898809 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 70.9

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

   5. Simplified 70.9%

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

   if -1e27 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.99997904554898809

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 95.4

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

   5. Simplified 95.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-cos.f64 89.7

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

   8. Simplified 89.7%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - \sin y \cdot z \leq -1 \cdot 10^{+27}:\\ \;\;\;\;x - \mathsf{fma}\left(y, z, -1\right)\\ \mathbf{elif}\;\left(x + \cos y\right) - \sin y \cdot z \leq 0.9999790455489881:\\ \;\;\;\;\cos y\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(y, z, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 5: 79.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sin y \cdot z\\ \mathbf{if}\;z \leq -8 \cdot 10^{+202}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 69000000000000:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (sin y) z))))
   (if (<= z -8e+202) t_0 (if (<= z 69000000000000.0) (+ x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -(sin(y) * z);
	double tmp;
	if (z <= -8e+202) {
		tmp = t_0;
	} else if (z <= 69000000000000.0) {
		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 <= (-8d+202)) then
        tmp = t_0
    else if (z <= 69000000000000.0d0) 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 <= -8e+202) {
		tmp = t_0;
	} else if (z <= 69000000000000.0) {
		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 <= -8e+202:
		tmp = t_0
	elif z <= 69000000000000.0:
		tmp = x + math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(sin(y) * z))
	tmp = 0.0
	if (z <= -8e+202)
		tmp = t_0;
	elseif (z <= 69000000000000.0)
		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 <= -8e+202)
		tmp = t_0;
	elseif (z <= 69000000000000.0)
		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, -8e+202], t$95$0, If[LessEqual[z, 69000000000000.0], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.9999999999999992e202 or 6.9e13 < 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. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 71.8

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

   5. Simplified 71.8%

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

   if -7.9999999999999992e202 < z < 6.9e13

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 89.4

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

   5. Simplified 89.4%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+202}:\\ \;\;\;\;-\sin y \cdot z\\ \mathbf{elif}\;z \leq 69000000000000:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;-\sin y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ \mathbf{if}\;y \leq -38:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.95:\\ \;\;\;\;\left(x + 1\right) - y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), z \cdot -0.16666666666666666\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y))))
   (if (<= y -38.0)
     t_0
     (if (<= y 1.95)
       (-
        (+ x 1.0)
        (*
         y
         (fma
          (* y y)
          (fma
           (* z (* y y))
           (fma (* y y) -0.0001984126984126984 0.008333333333333333)
           (* z -0.16666666666666666))
          z)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + cos(y);
	double tmp;
	if (y <= -38.0) {
		tmp = t_0;
	} else if (y <= 1.95) {
		tmp = (x + 1.0) - (y * fma((y * y), fma((z * (y * y)), fma((y * y), -0.0001984126984126984, 0.008333333333333333), (z * -0.16666666666666666)), z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + cos(y))
	tmp = 0.0
	if (y <= -38.0)
		tmp = t_0;
	elseif (y <= 1.95)
		tmp = Float64(Float64(x + 1.0) - Float64(y * fma(Float64(y * y), fma(Float64(z * Float64(y * y)), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), Float64(z * -0.16666666666666666)), z)));
	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, -38.0], t$95$0, If[LessEqual[y, 1.95], N[(N[(x + 1.0), $MachinePrecision] - N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -38 or 1.94999999999999996 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 55.7

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

   5. Simplified 55.7%

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

   if -38 < y < 1.94999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x + 1\right) - y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right) + z\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(x + 1\right) - y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right), z\right)} \]

   8. Simplified 98.9%

      \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), z \cdot -0.16666666666666666\right), z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -38:\\ \;\;\;\;x + \cos y\\ \mathbf{elif}\;y \leq 1.95:\\ \;\;\;\;\left(x + 1\right) - y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), z \cdot -0.16666666666666666\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+22}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+78}:\\ \;\;\;\;\left(x + 1\right) - y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), z \cdot -0.16666666666666666\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.4e+22)
   (+ x 1.0)
   (if (<= y 1.6e+78)
     (-
      (+ x 1.0)
      (*
       y
       (fma
        (* y y)
        (fma
         (* z (* y y))
         (fma (* y y) -0.0001984126984126984 0.008333333333333333)
         (* z -0.16666666666666666))
        z)))
     (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.4e+22) {
		tmp = x + 1.0;
	} else if (y <= 1.6e+78) {
		tmp = (x + 1.0) - (y * fma((y * y), fma((z * (y * y)), fma((y * y), -0.0001984126984126984, 0.008333333333333333), (z * -0.16666666666666666)), z));
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.4e+22)
		tmp = Float64(x + 1.0);
	elseif (y <= 1.6e+78)
		tmp = Float64(Float64(x + 1.0) - Float64(y * fma(Float64(y * y), fma(Float64(z * Float64(y * y)), fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333), Float64(z * -0.16666666666666666)), z)));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.4e+22], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 1.6e+78], N[(N[(x + 1.0), $MachinePrecision] - N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.4000000000000004e22 or 1.59999999999999997e78 < 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. lower-+.f64 31.8

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

   5. Simplified 31.8%

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

   if -5.4000000000000004e22 < y < 1.59999999999999997e78

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 96.5

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

   5. Simplified 96.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x + 1\right) - y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right) + z\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(x + 1\right) - y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right), z\right)} \]

   8. Simplified 90.0%

      \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), z \cdot -0.16666666666666666\right), z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+22}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+78}:\\ \;\;\;\;x - \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.4e+22)
   (+ x 1.0)
   (if (<= y 1.6e+78)
     (-
      x
      (fma
       z
       (fma
        (fma
         y
         (* y (fma (* y y) -0.0001984126984126984 0.008333333333333333))
         -0.16666666666666666)
        (* y (* y y))
        y)
       -1.0))
     (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.4e+22) {
		tmp = x + 1.0;
	} else if (y <= 1.6e+78) {
		tmp = x - fma(z, fma(fma(y, (y * fma((y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), (y * (y * y)), y), -1.0);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.4e+22)
		tmp = Float64(x + 1.0);
	elseif (y <= 1.6e+78)
		tmp = Float64(x - fma(z, fma(fma(y, Float64(y * fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), Float64(y * Float64(y * y)), y), -1.0));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.4e+22], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 1.6e+78], N[(x - N[(z * N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.4000000000000004e22 or 1.59999999999999997e78 < 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. lower-+.f64 31.8

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

   5. Simplified 31.8%

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

   if -5.4000000000000004e22 < y < 1.59999999999999997e78

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 96.5

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

   5. Simplified 96.5%

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

      1. lift-+.f64 N/A

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

      2. lift-sin.f64 N/A

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

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

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 96.5

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

   7. Applied egg-rr 96.5%

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

   8. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. sub-neg N/A

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

      6. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. /-rgt-identity N/A

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

      13. metadata-eval N/A

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

      14. rgt-mult-inverse N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. distribute-neg-frac2 N/A

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

      17. mul-1-neg N/A

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

      18. associate-*r/ N/A

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

      19. *-rgt-identity N/A

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

      20. times-frac N/A

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

      21. *-lft-identity N/A

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

   10. Simplified 96.5%

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

   11. Taylor expanded in y around 0

       \[\leadsto x - \mathsf{fma}\left(z, \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}, -1\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto x - \mathsf{fma}\left(z, y \cdot \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) + 1\right)}, -1\right) \]

       2. distribute-lft-in N/A

          \[\leadsto x - \mathsf{fma}\left(z, \color{blue}{y \cdot \left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right) + y \cdot 1}, -1\right) \]

       3. associate-*r* N/A

          \[\leadsto x - \mathsf{fma}\left(z, \color{blue}{\left(y \cdot {y}^{2}\right) \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)} + y \cdot 1, -1\right) \]

       4. *-commutative N/A

          \[\leadsto x - \mathsf{fma}\left(z, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot \left(y \cdot {y}^{2}\right)} + y \cdot 1, -1\right) \]

       5. *-rgt-identity N/A

          \[\leadsto x - \mathsf{fma}\left(z, \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot \left(y \cdot {y}^{2}\right) + \color{blue}{y}, -1\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto x - \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}, y \cdot {y}^{2}, y\right)}, -1\right) \]

   13. Simplified 90.0%

       \[\leadsto x - \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), y \cdot \left(y \cdot y\right), y\right)}, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 69.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+25}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+42}:\\ \;\;\;\;\left(x + 1\right) - y \cdot \mathsf{fma}\left(z \cdot \left(y \cdot y\right), \mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.3e+25)
   (+ x 1.0)
   (if (<= y 4.2e+42)
     (-
      (+ x 1.0)
      (*
       y
       (fma
        (* z (* y y))
        (fma y (* y 0.008333333333333333) -0.16666666666666666)
        z)))
     (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.3e+25) {
		tmp = x + 1.0;
	} else if (y <= 4.2e+42) {
		tmp = (x + 1.0) - (y * fma((z * (y * y)), fma(y, (y * 0.008333333333333333), -0.16666666666666666), z));
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.3e+25)
		tmp = Float64(x + 1.0);
	elseif (y <= 4.2e+42)
		tmp = Float64(Float64(x + 1.0) - Float64(y * fma(Float64(z * Float64(y * y)), fma(y, Float64(y * 0.008333333333333333), -0.16666666666666666), z)));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.3e+25], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 4.2e+42], N[(N[(x + 1.0), $MachinePrecision] - N[(y * N[(N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2999999999999998e25 or 4.19999999999999991e42 < 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. lower-+.f64 31.5

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

   5. Simplified 31.5%

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

   if -2.2999999999999998e25 < y < 4.19999999999999991e42

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 96.5

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

   5. Simplified 96.5%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. lower-fma.f64 N/A

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

   8. Simplified 90.5%

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

Alternative 10: 69.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+22}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 5.4:\\ \;\;\;\;1 + \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot 0.16666666666666666, -0.5\right), -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.5e+22)
   (+ x 1.0)
   (if (<= y 5.4)
     (+ 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 <= -5.5e+22) {
		tmp = x + 1.0;
	} else if (y <= 5.4) {
		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 <= -5.5e+22)
		tmp = Float64(x + 1.0);
	elseif (y <= 5.4)
		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, -5.5e+22], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 5.4], 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 < -5.50000000000000021e22 or 5.4000000000000004 < 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. +-commutative N/A

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

      2. lower-+.f64 30.6

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

   5. Simplified 30.6%

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

   if -5.50000000000000021e22 < y < 5.4000000000000004

   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. lower-+.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. lower-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. lower-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. lower-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. lower-*.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. lower-neg.f64 96.0

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

   5. Simplified 96.0%

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

Alternative 11: 69.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+22}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+78}:\\ \;\;\;\;1 + \mathsf{fma}\left(y \cdot z, \mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.5e+22)
   (+ x 1.0)
   (if (<= y 1.6e+78)
     (+ 1.0 (fma (* y z) (fma 0.16666666666666666 (* y y) -1.0) x))
     (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.5e+22) {
		tmp = x + 1.0;
	} else if (y <= 1.6e+78) {
		tmp = 1.0 + fma((y * z), fma(0.16666666666666666, (y * y), -1.0), x);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.5e+22)
		tmp = Float64(x + 1.0);
	elseif (y <= 1.6e+78)
		tmp = Float64(1.0 + fma(Float64(y * z), fma(0.16666666666666666, Float64(y * y), -1.0), x));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.5e+22], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 1.6e+78], N[(1.0 + N[(N[(y * z), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.50000000000000021e22 or 1.59999999999999997e78 < 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. lower-+.f64 31.8

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

   5. Simplified 31.8%

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

   if -5.50000000000000021e22 < y < 1.59999999999999997e78

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 96.5

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

   5. Simplified 96.5%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lft-identity N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 89.7

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

   8. Simplified 89.7%

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

Alternative 12: 69.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+57}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+43}:\\ \;\;\;\;x - \mathsf{fma}\left(y, z, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.5e+57)
   (+ x 1.0)
   (if (<= y 7.2e+43) (- x (fma y z -1.0)) (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.5e+57) {
		tmp = x + 1.0;
	} else if (y <= 7.2e+43) {
		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 <= -3.5e+57)
		tmp = Float64(x + 1.0);
	elseif (y <= 7.2e+43)
		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, -3.5e+57], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 7.2e+43], N[(x - N[(y * z + -1.0), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.4999999999999997e57 or 7.2000000000000002e43 < 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. lower-+.f64 30.0

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

   5. Simplified 30.0%

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

   if -3.4999999999999997e57 < y < 7.2000000000000002e43

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 88.6

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

   5. Simplified 88.6%

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

Alternative 13: 62.5% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+203}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+258}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- z))))
   (if (<= z -9.5e+203) t_0 (if (<= z 3.4e+258) (+ x 1.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * -z;
	double tmp;
	if (z <= -9.5e+203) {
		tmp = t_0;
	} else if (z <= 3.4e+258) {
		tmp = x + 1.0;
	} 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 = y * -z
    if (z <= (-9.5d+203)) then
        tmp = t_0
    else if (z <= 3.4d+258) then
        tmp = x + 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * -z;
	double tmp;
	if (z <= -9.5e+203) {
		tmp = t_0;
	} else if (z <= 3.4e+258) {
		tmp = x + 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * -z
	tmp = 0
	if z <= -9.5e+203:
		tmp = t_0
	elif z <= 3.4e+258:
		tmp = x + 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(-z))
	tmp = 0.0
	if (z <= -9.5e+203)
		tmp = t_0;
	elseif (z <= 3.4e+258)
		tmp = Float64(x + 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * -z;
	tmp = 0.0;
	if (z <= -9.5e+203)
		tmp = t_0;
	elseif (z <= 3.4e+258)
		tmp = x + 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * (-z)), $MachinePrecision]}, If[LessEqual[z, -9.5e+203], t$95$0, If[LessEqual[z, 3.4e+258], N[(x + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.4999999999999995e203 or 3.39999999999999981e258 < z

   1. Initial program 99.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. flip-- N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 99.6

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

      12. lift--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate--l+ N/A

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{-1}{z \cdot \sin y}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{-1}{z \cdot \sin y}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{-1}{\color{blue}{z \cdot \sin y}}} \]

      3. lower-sin.f64 86.5

         \[\leadsto \frac{1}{\frac{-1}{z \cdot \color{blue}{\sin y}}} \]

   7. Simplified 86.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{-1}{z \cdot \sin y}}} \]

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 39.0

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

   10. Simplified 39.0%

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

   if -9.4999999999999995e203 < z < 3.39999999999999981e258

   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. lower-+.f64 65.8

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

   5. Simplified 65.8%

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

Alternative 14: 61.4% 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. lower-+.f64 56.9

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

5. Simplified 56.9%

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

Alternative 15: 21.1% 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. lower-+.f64 56.9

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

5. Simplified 56.9%

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

6. Taylor expanded in x around 0

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

8. Simplified 23.1%

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

Reproduce

herbie shell --seed 2024207 
(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))))