Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3

Percentage Accurate: 88.2% → 99.8%

Time: 7.8s

Alternatives: 13

Speedup: 0.6×

• 20.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + (y * (z - x))) / 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 + (y * (z - x))) / z
end function

Java

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

Python

def code(x, y, z):
	return (x + (y * (z - x))) / z

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + (y * (z - x))) / 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 + (y * (z - x))) / z
end function

Java

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

Python

def code(x, y, z):
	return (x + (y * (z - x))) / z

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ y - {\left(\frac{\frac{z}{x}}{y + -1}\right)}^{-1} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- y (pow (/ (/ z x) (+ y -1.0)) -1.0)))

C

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

Java

public static double code(double x, double y, double z) {
	return y - Math.pow(((z / x) / (y + -1.0)), -1.0);
}

Python

def code(x, y, z):
	return y - math.pow(((z / x) / (y + -1.0)), -1.0)

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(y - N[Power[N[(N[(z / x), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.7%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in y around 0 95.6%

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

4. Taylor expanded in x around 0 96.1%

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

   1. distribute-lft-in 93.0%

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

   2. mul-1-neg 93.0%

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

   3. distribute-rgt-neg-in 93.0%

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

   4. associate-/l* 93.4%

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

   5. mul-1-neg 93.4%

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

   6. associate-*r/ 93.5%

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

   7. *-rgt-identity 93.5%

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

   8. remove-double-neg 93.5%

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

   9. distribute-frac-neg2 93.5%

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

   10. unsub-neg 93.5%

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

   11. mul-1-neg 93.5%

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

   12. distribute-frac-neg2 93.5%

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

   13. div-sub 96.7%

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

   14. unsub-neg 96.7%

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

   15. mul-1-neg 96.7%

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

   16. +-commutative 96.7%

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

   17. distribute-neg-frac2 96.7%

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

   18. unsub-neg 96.7%

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

6. Simplified 96.7%

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

   1. clear-num 96.5%

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

   2. inv-pow 96.5%

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

   3. associate-/r* 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

   \[\leadsto y - {\left(\frac{\frac{z}{x}}{y + -1}\right)}^{-1} \]
10. Add Preprocessing

Alternative 2: 75.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-169} \lor \neg \left(z \leq 3 \cdot 10^{-293} \lor \neg \left(z \leq 6.2 \cdot 10^{-208}\right) \land z \leq 8.6 \cdot 10^{-121}\right):\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{-x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.4e-169)
         (not (or (<= z 3e-293) (and (not (<= z 6.2e-208)) (<= z 8.6e-121)))))
   (+ y (/ x z))
   (/ y (/ z (- x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.4e-169) || !((z <= 3e-293) || (!(z <= 6.2e-208) && (z <= 8.6e-121)))) {
		tmp = y + (x / z);
	} else {
		tmp = y / (z / -x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.4d-169)) .or. (.not. (z <= 3d-293) .or. (.not. (z <= 6.2d-208)) .and. (z <= 8.6d-121))) then
        tmp = y + (x / z)
    else
        tmp = y / (z / -x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.4e-169) || !((z <= 3e-293) || (!(z <= 6.2e-208) && (z <= 8.6e-121)))) {
		tmp = y + (x / z);
	} else {
		tmp = y / (z / -x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.4e-169) or not ((z <= 3e-293) or (not (z <= 6.2e-208) and (z <= 8.6e-121))):
		tmp = y + (x / z)
	else:
		tmp = y / (z / -x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.4e-169) || !((z <= 3e-293) || (!(z <= 6.2e-208) && (z <= 8.6e-121))))
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y / Float64(z / Float64(-x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.4e-169) || ~(((z <= 3e-293) || (~((z <= 6.2e-208)) && (z <= 8.6e-121)))))
		tmp = y + (x / z);
	else
		tmp = y / (z / -x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.4e-169], N[Not[Or[LessEqual[z, 3e-293], And[N[Not[LessEqual[z, 6.2e-208]], $MachinePrecision], LessEqual[z, 8.6e-121]]]], $MachinePrecision]], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(z / (-x)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.39999999999999994e-169 or 3.0000000000000002e-293 < z < 6.1999999999999996e-208 or 8.5999999999999993e-121 < z

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.4%

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

   4. Taylor expanded in x around 0 96.9%

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

      1. distribute-lft-in 95.9%

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

      2. mul-1-neg 95.9%

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

      3. distribute-rgt-neg-in 95.9%

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

      4. associate-/l* 94.6%

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

      5. mul-1-neg 94.6%

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

      6. associate-*r/ 94.7%

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

      7. *-rgt-identity 94.7%

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

      8. remove-double-neg 94.7%

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

      9. distribute-frac-neg2 94.7%

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

      10. unsub-neg 94.7%

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

      11. mul-1-neg 94.7%

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

      12. distribute-frac-neg2 94.7%

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

      13. div-sub 95.7%

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

      14. unsub-neg 95.7%

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

      15. mul-1-neg 95.7%

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

      16. +-commutative 95.7%

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

      17. distribute-neg-frac2 95.7%

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

      18. unsub-neg 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in y around 0 79.5%

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

      1. associate-*r/ 79.5%

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

      2. neg-mul-1 79.5%

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

   9. Simplified 79.5%

      \[\leadsto y - \color{blue}{\frac{-x}{z}} \]
   10. Step-by-step derivation

       1. sub-neg 79.5%

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

       2. distribute-frac-neg 79.5%

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

       3. remove-double-neg 79.5%

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

       4. +-commutative 79.5%

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

   11. Applied egg-rr 79.5%

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

   if -1.39999999999999994e-169 < z < 3.0000000000000002e-293 or 6.1999999999999996e-208 < z < 8.5999999999999993e-121

   1. Initial program 99.8%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 78.4%

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

      1. associate-/l* 80.0%

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

      2. div-sub 80.1%

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

      3. sub-neg 80.1%

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

      4. *-inverses 80.1%

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

      5. sub-neg 80.1%

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

   5. Simplified 80.1%

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

   6. Taylor expanded in x around inf 71.4%

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

      1. associate-*r/ 71.4%

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

      2. mul-1-neg 71.4%

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

      3. distribute-rgt-neg-in 71.4%

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

      4. associate-*r/ 66.7%

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

   8. Simplified 66.7%

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

      1. distribute-frac-neg 66.7%

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

      2. distribute-rgt-neg-out 66.7%

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

      3. associate-/l* 71.4%

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

      4. *-commutative 71.4%

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

      5. associate-/l* 73.0%

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

   10. Applied egg-rr 73.0%

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

       1. clear-num 73.0%

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

       2. un-div-inv 73.0%

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

   12. Applied egg-rr 73.0%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-169} \lor \neg \left(z \leq 3 \cdot 10^{-293} \lor \neg \left(z \leq 6.2 \cdot 10^{-208}\right) \land z \leq 8.6 \cdot 10^{-121}\right):\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{-x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-171} \lor \neg \left(z \leq 1.4 \cdot 10^{-293} \lor \neg \left(z \leq 2.3 \cdot 10^{-210}\right) \land z \leq 2.25 \cdot 10^{-119}\right):\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.6e-171)
         (not
          (or (<= z 1.4e-293) (and (not (<= z 2.3e-210)) (<= z 2.25e-119)))))
   (+ y (/ x z))
   (* y (/ x (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.6e-171) || !((z <= 1.4e-293) || (!(z <= 2.3e-210) && (z <= 2.25e-119)))) {
		tmp = y + (x / z);
	} else {
		tmp = y * (x / -z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2.6d-171)) .or. (.not. (z <= 1.4d-293) .or. (.not. (z <= 2.3d-210)) .and. (z <= 2.25d-119))) then
        tmp = y + (x / z)
    else
        tmp = y * (x / -z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.6e-171) || !((z <= 1.4e-293) || (!(z <= 2.3e-210) && (z <= 2.25e-119)))) {
		tmp = y + (x / z);
	} else {
		tmp = y * (x / -z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.6e-171) or not ((z <= 1.4e-293) or (not (z <= 2.3e-210) and (z <= 2.25e-119))):
		tmp = y + (x / z)
	else:
		tmp = y * (x / -z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.6e-171) || !((z <= 1.4e-293) || (!(z <= 2.3e-210) && (z <= 2.25e-119))))
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y * Float64(x / Float64(-z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.6e-171) || ~(((z <= 1.4e-293) || (~((z <= 2.3e-210)) && (z <= 2.25e-119)))))
		tmp = y + (x / z);
	else
		tmp = y * (x / -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.6e-171], N[Not[Or[LessEqual[z, 1.4e-293], And[N[Not[LessEqual[z, 2.3e-210]], $MachinePrecision], LessEqual[z, 2.25e-119]]]], $MachinePrecision]], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / (-z)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.60000000000000005e-171 or 1.40000000000000013e-293 < z < 2.3e-210 or 2.2500000000000001e-119 < z

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.4%

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

   4. Taylor expanded in x around 0 96.9%

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

      1. distribute-lft-in 95.9%

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

      2. mul-1-neg 95.9%

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

      3. distribute-rgt-neg-in 95.9%

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

      4. associate-/l* 94.6%

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

      5. mul-1-neg 94.6%

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

      6. associate-*r/ 94.7%

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

      7. *-rgt-identity 94.7%

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

      8. remove-double-neg 94.7%

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

      9. distribute-frac-neg2 94.7%

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

      10. unsub-neg 94.7%

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

      11. mul-1-neg 94.7%

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

      12. distribute-frac-neg2 94.7%

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

      13. div-sub 95.7%

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

      14. unsub-neg 95.7%

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

      15. mul-1-neg 95.7%

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

      16. +-commutative 95.7%

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

      17. distribute-neg-frac2 95.7%

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

      18. unsub-neg 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in y around 0 79.5%

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

      1. associate-*r/ 79.5%

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

      2. neg-mul-1 79.5%

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

   9. Simplified 79.5%

      \[\leadsto y - \color{blue}{\frac{-x}{z}} \]
   10. Step-by-step derivation

       1. sub-neg 79.5%

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

       2. distribute-frac-neg 79.5%

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

       3. remove-double-neg 79.5%

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

       4. +-commutative 79.5%

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

   11. Applied egg-rr 79.5%

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

   if -2.60000000000000005e-171 < z < 1.40000000000000013e-293 or 2.3e-210 < z < 2.2500000000000001e-119

   1. Initial program 99.8%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 78.4%

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

      1. associate-/l* 80.0%

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

      2. div-sub 80.1%

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

      3. sub-neg 80.1%

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

      4. *-inverses 80.1%

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

      5. sub-neg 80.1%

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

   5. Simplified 80.1%

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

   6. Taylor expanded in x around inf 71.4%

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

      1. associate-*r/ 71.4%

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

      2. mul-1-neg 71.4%

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

      3. distribute-rgt-neg-in 71.4%

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

      4. associate-*r/ 66.7%

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

   8. Simplified 66.7%

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

      1. distribute-frac-neg 66.7%

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

      2. distribute-rgt-neg-out 66.7%

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

      3. associate-/l* 71.4%

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

      4. *-commutative 71.4%

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

      5. associate-/l* 73.0%

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

   10. Applied egg-rr 73.0%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-171} \lor \neg \left(z \leq 1.4 \cdot 10^{-293} \lor \neg \left(z \leq 2.3 \cdot 10^{-210}\right) \land z \leq 2.25 \cdot 10^{-119}\right):\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \frac{x}{z}\\ \mathbf{if}\;z \leq -1.02 \cdot 10^{-169}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-293}:\\ \;\;\;\;\frac{y}{\frac{z}{-x}}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-208} \lor \neg \left(z \leq 8 \cdot 10^{-121}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (/ x z))))
   (if (<= z -1.02e-169)
     t_0
     (if (<= z 3.8e-293)
       (/ y (/ z (- x)))
       (if (or (<= z 3.1e-208) (not (<= z 8e-121))) t_0 (* x (- (/ y z))))))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x / z);
	double tmp;
	if (z <= -1.02e-169) {
		tmp = t_0;
	} else if (z <= 3.8e-293) {
		tmp = y / (z / -x);
	} else if ((z <= 3.1e-208) || !(z <= 8e-121)) {
		tmp = t_0;
	} else {
		tmp = x * -(y / z);
	}
	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 + (x / z)
    if (z <= (-1.02d-169)) then
        tmp = t_0
    else if (z <= 3.8d-293) then
        tmp = y / (z / -x)
    else if ((z <= 3.1d-208) .or. (.not. (z <= 8d-121))) then
        tmp = t_0
    else
        tmp = x * -(y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y + (x / z);
	double tmp;
	if (z <= -1.02e-169) {
		tmp = t_0;
	} else if (z <= 3.8e-293) {
		tmp = y / (z / -x);
	} else if ((z <= 3.1e-208) || !(z <= 8e-121)) {
		tmp = t_0;
	} else {
		tmp = x * -(y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y + (x / z)
	tmp = 0
	if z <= -1.02e-169:
		tmp = t_0
	elif z <= 3.8e-293:
		tmp = y / (z / -x)
	elif (z <= 3.1e-208) or not (z <= 8e-121):
		tmp = t_0
	else:
		tmp = x * -(y / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x / z))
	tmp = 0.0
	if (z <= -1.02e-169)
		tmp = t_0;
	elseif (z <= 3.8e-293)
		tmp = Float64(y / Float64(z / Float64(-x)));
	elseif ((z <= 3.1e-208) || !(z <= 8e-121))
		tmp = t_0;
	else
		tmp = Float64(x * Float64(-Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y + (x / z);
	tmp = 0.0;
	if (z <= -1.02e-169)
		tmp = t_0;
	elseif (z <= 3.8e-293)
		tmp = y / (z / -x);
	elseif ((z <= 3.1e-208) || ~((z <= 8e-121)))
		tmp = t_0;
	else
		tmp = x * -(y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.02e-169], t$95$0, If[LessEqual[z, 3.8e-293], N[(y / N[(z / (-x)), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 3.1e-208], N[Not[LessEqual[z, 8e-121]], $MachinePrecision]], t$95$0, N[(x * (-N[(y / z), $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.01999999999999996e-169 or 3.8e-293 < z < 3.0999999999999998e-208 or 7.9999999999999998e-121 < z

   1. Initial program 88.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.4%

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

   4. Taylor expanded in x around 0 96.9%

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

      1. distribute-lft-in 95.9%

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

      2. mul-1-neg 95.9%

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

      3. distribute-rgt-neg-in 95.9%

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

      4. associate-/l* 94.6%

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

      5. mul-1-neg 94.6%

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

      6. associate-*r/ 94.7%

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

      7. *-rgt-identity 94.7%

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

      8. remove-double-neg 94.7%

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

      9. distribute-frac-neg2 94.7%

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

      10. unsub-neg 94.7%

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

      11. mul-1-neg 94.7%

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

      12. distribute-frac-neg2 94.7%

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

      13. div-sub 95.7%

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

      14. unsub-neg 95.7%

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

      15. mul-1-neg 95.7%

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

      16. +-commutative 95.7%

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

      17. distribute-neg-frac2 95.7%

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

      18. unsub-neg 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in y around 0 79.5%

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

      1. associate-*r/ 79.5%

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

      2. neg-mul-1 79.5%

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

   9. Simplified 79.5%

      \[\leadsto y - \color{blue}{\frac{-x}{z}} \]
   10. Step-by-step derivation

       1. sub-neg 79.5%

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

       2. distribute-frac-neg 79.5%

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

       3. remove-double-neg 79.5%

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

       4. +-commutative 79.5%

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

   11. Applied egg-rr 79.5%

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

   if -1.01999999999999996e-169 < z < 3.8e-293

   1. Initial program 99.9%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 75.7%

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

      1. associate-/l* 78.3%

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

      2. div-sub 78.3%

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

      3. sub-neg 78.3%

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

      4. *-inverses 78.3%

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

      5. sub-neg 78.3%

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

   5. Simplified 78.3%

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

   6. Taylor expanded in x around inf 69.7%

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

      1. associate-*r/ 69.7%

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

      2. mul-1-neg 69.7%

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

      3. distribute-rgt-neg-in 69.7%

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

      4. associate-*r/ 62.3%

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

   8. Simplified 62.3%

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

      1. distribute-frac-neg 62.3%

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

      2. distribute-rgt-neg-out 62.3%

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

      3. associate-/l* 69.7%

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

      4. *-commutative 69.7%

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

      5. associate-/l* 72.3%

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

   10. Applied egg-rr 72.3%

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

       1. clear-num 72.3%

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

       2. un-div-inv 72.3%

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

   12. Applied egg-rr 72.3%

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

   if 3.0999999999999998e-208 < z < 7.9999999999999998e-121

   1. Initial program 99.8%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 83.0%

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

      1. associate-/l* 83.1%

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

      2. div-sub 83.1%

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

      3. sub-neg 83.1%

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

      4. *-inverses 83.1%

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

      5. sub-neg 83.1%

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

   5. Simplified 83.1%

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

   6. Taylor expanded in x around inf 74.1%

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

      1. associate-*r/ 74.1%

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

      2. mul-1-neg 74.1%

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

      3. distribute-rgt-neg-in 74.1%

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

      4. associate-*r/ 74.3%

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

   8. Simplified 74.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{-169}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-293}:\\ \;\;\;\;\frac{y}{\frac{z}{-x}}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-208} \lor \neg \left(z \leq 8 \cdot 10^{-121}\right):\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\frac{y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -122 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -122.0) (not (<= y 1.0))) (* y (- 1.0 (/ x z))) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -122.0) || !(y <= 1.0)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (x / z);
	}
	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 ((y <= (-122.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * (1.0d0 - (x / z))
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -122.0) || !(y <= 1.0)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -122.0) or not (y <= 1.0):
		tmp = y * (1.0 - (x / z))
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -122.0) || !(y <= 1.0))
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -122.0) || ~((y <= 1.0)))
		tmp = y * (1.0 - (x / z));
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -122.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -122 or 1 < y

   1. Initial program 82.6%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 81.6%

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

      1. associate-/l* 98.8%

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

      2. div-sub 98.8%

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

      3. sub-neg 98.8%

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

      4. *-inverses 98.8%

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

      5. sub-neg 98.8%

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

   5. Simplified 98.8%

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

   if -122 < y < 1

   1. Initial program 99.9%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 95.8%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. distribute-lft-in 98.1%

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

      2. mul-1-neg 98.1%

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

      3. distribute-rgt-neg-in 98.1%

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

      4. associate-/l* 98.1%

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

      5. mul-1-neg 98.1%

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

      6. associate-*r/ 98.3%

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

      7. *-rgt-identity 98.3%

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

      8. remove-double-neg 98.3%

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

      9. distribute-frac-neg2 98.3%

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

      10. unsub-neg 98.3%

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

      11. mul-1-neg 98.3%

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

      12. distribute-frac-neg2 98.3%

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

      13. div-sub 100.0%

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

      14. unsub-neg 100.0%

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

      15. mul-1-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. distribute-neg-frac2 100.0%

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

      18. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 97.4%

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

      1. associate-*r/ 97.4%

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

      2. neg-mul-1 97.4%

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

   9. Simplified 97.4%

      \[\leadsto y - \color{blue}{\frac{-x}{z}} \]
   10. Step-by-step derivation

       1. sub-neg 97.4%

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

       2. distribute-frac-neg 97.4%

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

       3. remove-double-neg 97.4%

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

       4. +-commutative 97.4%

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

   11. Applied egg-rr 97.4%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -122 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+66} \lor \neg \left(x \leq 9.8 \cdot 10^{-71}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.4e+66) (not (<= x 9.8e-71)))
   (* x (/ (- 1.0 y) z))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.4e+66) || !(x <= 9.8e-71)) {
		tmp = x * ((1.0 - y) / z);
	} else {
		tmp = y + (x / z);
	}
	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 <= (-2.4d+66)) .or. (.not. (x <= 9.8d-71))) then
        tmp = x * ((1.0d0 - y) / z)
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.4e+66) || !(x <= 9.8e-71)) {
		tmp = x * ((1.0 - y) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.4e+66) or not (x <= 9.8e-71):
		tmp = x * ((1.0 - y) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.4e+66) || !(x <= 9.8e-71))
		tmp = Float64(x * Float64(Float64(1.0 - y) / z));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.4e+66) || ~((x <= 9.8e-71)))
		tmp = x * ((1.0 - y) / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.4e+66], N[Not[LessEqual[x, 9.8e-71]], $MachinePrecision]], N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4000000000000002e66 or 9.7999999999999994e-71 < x

   1. Initial program 93.2%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 81.1%

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

      1. associate-/l* 83.6%

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

      2. mul-1-neg 83.6%

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

      3. unsub-neg 83.6%

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

   5. Simplified 83.6%

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

   if -2.4000000000000002e66 < x < 9.7999999999999994e-71

   1. Initial program 87.8%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.9%

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

   4. Taylor expanded in x around 0 92.6%

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

      1. distribute-lft-in 92.5%

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

      2. mul-1-neg 92.5%

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

      3. distribute-rgt-neg-in 92.5%

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

      4. associate-/l* 99.2%

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

      5. mul-1-neg 99.2%

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

      6. associate-*r/ 99.3%

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

      7. *-rgt-identity 99.3%

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

      8. remove-double-neg 99.3%

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

      9. distribute-frac-neg2 99.3%

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

      10. unsub-neg 99.3%

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

      11. mul-1-neg 99.3%

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

      12. distribute-frac-neg2 99.3%

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

      13. div-sub 99.3%

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

      14. unsub-neg 99.3%

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

      15. mul-1-neg 99.3%

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

      16. +-commutative 99.3%

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

      17. distribute-neg-frac2 99.3%

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

      18. unsub-neg 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in y around 0 85.8%

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

      1. associate-*r/ 85.8%

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

      2. neg-mul-1 85.8%

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

   9. Simplified 85.8%

      \[\leadsto y - \color{blue}{\frac{-x}{z}} \]
   10. Step-by-step derivation

       1. sub-neg 85.8%

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

       2. distribute-frac-neg 85.8%

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

       3. remove-double-neg 85.8%

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

       4. +-commutative 85.8%

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

   11. Applied egg-rr 85.8%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+66} \lor \neg \left(x \leq 9.8 \cdot 10^{-71}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-15}:\\ \;\;\;\;y - \frac{x \cdot \left(y + -1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2e-15)
   (- y (/ (* x (+ y -1.0)) z))
   (+ (/ x z) (* y (- 1.0 (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2e-15) {
		tmp = y - ((x * (y + -1.0)) / z);
	} else {
		tmp = (x / z) + (y * (1.0 - (x / z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 2d-15) then
        tmp = y - ((x * (y + (-1.0d0))) / z)
    else
        tmp = (x / z) + (y * (1.0d0 - (x / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2e-15) {
		tmp = y - ((x * (y + -1.0)) / z);
	} else {
		tmp = (x / z) + (y * (1.0 - (x / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2e-15:
		tmp = y - ((x * (y + -1.0)) / z)
	else:
		tmp = (x / z) + (y * (1.0 - (x / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2e-15)
		tmp = Float64(y - Float64(Float64(x * Float64(y + -1.0)) / z));
	else
		tmp = Float64(Float64(x / z) + Float64(y * Float64(1.0 - Float64(x / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2e-15)
		tmp = y - ((x * (y + -1.0)) / z);
	else
		tmp = (x / z) + (y * (1.0 - (x / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2e-15], N[(y - N[(N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] + N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.0000000000000002e-15

   1. Initial program 94.7%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 94.0%

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

   4. Taylor expanded in x around 0 94.7%

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

      1. distribute-lft-in 90.3%

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

      2. mul-1-neg 90.3%

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

      3. distribute-rgt-neg-in 90.3%

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

      4. associate-/l* 94.4%

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

      5. mul-1-neg 94.4%

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

      6. associate-*r/ 94.5%

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

      7. *-rgt-identity 94.5%

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

      8. remove-double-neg 94.5%

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

      9. distribute-frac-neg2 94.5%

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

      10. unsub-neg 94.5%

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

      11. mul-1-neg 94.5%

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

      12. distribute-frac-neg2 94.5%

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

      13. div-sub 98.8%

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

      14. unsub-neg 98.8%

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

      15. mul-1-neg 98.8%

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

      16. +-commutative 98.8%

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

      17. distribute-neg-frac2 98.8%

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

      18. unsub-neg 98.8%

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

   6. Simplified 98.8%

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

   if 2.0000000000000002e-15 < z

   1. Initial program 80.5%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-15}:\\ \;\;\;\;y - \frac{x \cdot \left(y + -1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \left(\frac{1 - y}{z} + \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x \cdot \left(y + -1\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2e+36)
   (* x (+ (/ (- 1.0 y) z) (/ y x)))
   (- y (/ (* x (+ y -1.0)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e+36) {
		tmp = x * (((1.0 - y) / z) + (y / x));
	} else {
		tmp = y - ((x * (y + -1.0)) / z);
	}
	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 <= (-2d+36)) then
        tmp = x * (((1.0d0 - y) / z) + (y / x))
    else
        tmp = y - ((x * (y + (-1.0d0))) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e+36) {
		tmp = x * (((1.0 - y) / z) + (y / x));
	} else {
		tmp = y - ((x * (y + -1.0)) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2e+36:
		tmp = x * (((1.0 - y) / z) + (y / x))
	else:
		tmp = y - ((x * (y + -1.0)) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2e+36)
		tmp = Float64(x * Float64(Float64(Float64(1.0 - y) / z) + Float64(y / x)));
	else
		tmp = Float64(y - Float64(Float64(x * Float64(y + -1.0)) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2e+36)
		tmp = x * (((1.0 - y) / z) + (y / x));
	else
		tmp = y - ((x * (y + -1.0)) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2e+36], N[(x * N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y - N[(N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000008e36

   1. Initial program 88.9%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.7%

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

      1. associate-+r+ 99.7%

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

      2. +-commutative 99.7%

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

      3. mul-1-neg 99.7%

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

      4. unsub-neg 99.7%

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

      5. div-sub 99.7%

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

   5. Simplified 99.7%

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

   if -2.00000000000000008e36 < x

   1. Initial program 91.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.3%

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

   4. Taylor expanded in x around 0 94.8%

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

      1. distribute-lft-in 93.2%

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

      2. mul-1-neg 93.2%

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

      3. distribute-rgt-neg-in 93.2%

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

      4. associate-/l* 96.3%

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

      5. mul-1-neg 96.3%

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

      6. associate-*r/ 96.4%

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

      7. *-rgt-identity 96.4%

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

      8. remove-double-neg 96.4%

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

      9. distribute-frac-neg2 96.4%

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

      10. unsub-neg 96.4%

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

      11. mul-1-neg 96.4%

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

      12. distribute-frac-neg2 96.4%

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

      13. div-sub 98.0%

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

      14. unsub-neg 98.0%

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

      15. mul-1-neg 98.0%

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

      16. +-commutative 98.0%

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

      17. distribute-neg-frac2 98.0%

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

      18. unsub-neg 98.0%

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

   6. Simplified 98.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \left(\frac{1 - y}{z} + \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x \cdot \left(y + -1\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-51} \lor \neg \left(y \leq 6.5 \cdot 10^{-24}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.3e-51) (not (<= y 6.5e-24))) (* z (/ y z)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.3e-51) || !(y <= 6.5e-24)) {
		tmp = z * (y / z);
	} else {
		tmp = x / z;
	}
	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 ((y <= (-2.3d-51)) .or. (.not. (y <= 6.5d-24))) then
        tmp = z * (y / z)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.3e-51) || !(y <= 6.5e-24)) {
		tmp = z * (y / z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.3e-51) or not (y <= 6.5e-24):
		tmp = z * (y / z)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.3e-51) || !(y <= 6.5e-24))
		tmp = Float64(z * Float64(y / z));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.3e-51) || ~((y <= 6.5e-24)))
		tmp = z * (y / z);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.3e-51], N[Not[LessEqual[y, 6.5e-24]], $MachinePrecision]], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.30000000000000002e-51 or 6.5e-24 < y

   1. Initial program 84.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 34.1%

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

      1. *-commutative 34.1%

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

      2. associate-/l* 51.1%

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

   5. Applied egg-rr 51.1%

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

   if -2.30000000000000002e-51 < y < 6.5e-24

   1. Initial program 100.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 76.2%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-51} \lor \neg \left(y \leq 6.5 \cdot 10^{-24}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+15}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x \cdot \left(y + -1\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.5e+15) (* y (- 1.0 (/ x z))) (- y (/ (* x (+ y -1.0)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.5e+15) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y - ((x * (y + -1.0)) / z);
	}
	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 (y <= (-5.5d+15)) then
        tmp = y * (1.0d0 - (x / z))
    else
        tmp = y - ((x * (y + (-1.0d0))) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.5e+15) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y - ((x * (y + -1.0)) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5.5e+15:
		tmp = y * (1.0 - (x / z))
	else:
		tmp = y - ((x * (y + -1.0)) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.5e+15)
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	else
		tmp = Float64(y - Float64(Float64(x * Float64(y + -1.0)) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5.5e+15)
		tmp = y * (1.0 - (x / z));
	else
		tmp = y - ((x * (y + -1.0)) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.5e+15], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y - N[(N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5e15

   1. Initial program 83.2%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 83.2%

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

      1. associate-/l* 99.8%

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

      2. div-sub 99.8%

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

      3. sub-neg 99.8%

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

      4. *-inverses 99.8%

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

      5. sub-neg 99.8%

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

   5. Simplified 99.8%

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

   if -5.5e15 < y

   1. Initial program 93.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 94.1%

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

   4. Taylor expanded in x around 0 97.8%

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

      1. distribute-lft-in 93.6%

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

      2. mul-1-neg 93.6%

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

      3. distribute-rgt-neg-in 93.6%

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

      4. associate-/l* 93.6%

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

      5. mul-1-neg 93.6%

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

      6. associate-*r/ 93.7%

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

      7. *-rgt-identity 93.7%

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

      8. remove-double-neg 93.7%

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

      9. distribute-frac-neg2 93.7%

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

      10. unsub-neg 93.7%

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

      11. mul-1-neg 93.7%

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

      12. distribute-frac-neg2 93.7%

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

      13. div-sub 97.9%

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

      14. unsub-neg 97.9%

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

      15. mul-1-neg 97.9%

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

      16. +-commutative 97.9%

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

      17. distribute-neg-frac2 97.9%

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

      18. unsub-neg 97.9%

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

   6. Simplified 97.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+15}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x \cdot \left(y + -1\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-105}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-105) y (if (<= y 5e-23) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-105) {
		tmp = y;
	} else if (y <= 5e-23) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1d-105)) then
        tmp = y
    else if (y <= 5d-23) then
        tmp = x / z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-105) {
		tmp = y;
	} else if (y <= 5e-23) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1e-105:
		tmp = y
	elif y <= 5e-23:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-105)
		tmp = y;
	elseif (y <= 5e-23)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1e-105)
		tmp = y;
	elseif (y <= 5e-23)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e-105], y, If[LessEqual[y, 5e-23], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999965e-106 or 5.0000000000000002e-23 < y

   1. Initial program 85.6%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 46.5%

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

   if -9.99999999999999965e-106 < y < 5.0000000000000002e-23

   1. Initial program 100.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 80.2%

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

Alternative 12: 78.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.7%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in y around 0 95.6%

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

4. Taylor expanded in x around 0 96.1%

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

   1. distribute-lft-in 93.0%

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

   2. mul-1-neg 93.0%

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

   3. distribute-rgt-neg-in 93.0%

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

   4. associate-/l* 93.4%

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

   5. mul-1-neg 93.4%

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

   6. associate-*r/ 93.5%

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

   7. *-rgt-identity 93.5%

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

   8. remove-double-neg 93.5%

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

   9. distribute-frac-neg2 93.5%

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

   10. unsub-neg 93.5%

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

   11. mul-1-neg 93.5%

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

   12. distribute-frac-neg2 93.5%

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

   13. div-sub 96.7%

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

   14. unsub-neg 96.7%

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

   15. mul-1-neg 96.7%

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

   16. +-commutative 96.7%

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

   17. distribute-neg-frac2 96.7%

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

   18. unsub-neg 96.7%

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

6. Simplified 96.7%

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

7. Taylor expanded in y around 0 71.7%

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

   1. associate-*r/ 71.7%

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

   2. neg-mul-1 71.7%

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

9. Simplified 71.7%

   \[\leadsto y - \color{blue}{\frac{-x}{z}} \]
10. Step-by-step derivation

    1. sub-neg 71.7%

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

    2. distribute-frac-neg 71.7%

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

    3. remove-double-neg 71.7%

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

    4. +-commutative 71.7%

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

11. Applied egg-rr 71.7%

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

12. Final simplification 71.7%

    \[\leadsto y + \frac{x}{z} \]
13. Add Preprocessing

Alternative 13: 41.9% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 90.7%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 37.9%

   \[\leadsto \color{blue}{y} \]
4. Add Preprocessing

Developer target: 94.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (y + (x / z)) - (y / (z / x));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024085 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :alt
  (- (+ y (/ x z)) (/ y (/ z x)))

  (/ (+ x (* y (- z x))) z))