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

Percentage Accurate: 87.6% → 99.9%

Time: 8.2s

Alternatives: 9

Speedup: 0.8×

• 20.9% 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: 87.6% 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: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ y + \frac{\frac{x}{z}}{\frac{1}{1 - y}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

2. Taylor expanded in x around 0 96.1%

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

   1. +-commutative 96.1%

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

   2. mul-1-neg 96.1%

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

   3. unsub-neg 96.1%

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

   4. div-sub 96.1%

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

   5. clear-num 96.1%

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

   6. associate-/r/ 96.4%

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

   7. clear-num 96.5%

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

   8. div-inv 96.5%

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

   9. associate-/r* 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 74.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+209} \lor \neg \left(y \leq -1.65 \cdot 10^{+154}\right) \land \left(y \leq -1.19 \cdot 10^{+82} \lor \neg \left(y \leq 60000000000000\right)\right):\\ \;\;\;\;-\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.5e+209)
         (and (not (<= y -1.65e+154))
              (or (<= y -1.19e+82) (not (<= y 60000000000000.0)))))
   (- (/ y (/ z x)))
   (+ (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.5e+209) || (!(y <= -1.65e+154) && ((y <= -1.19e+82) || !(y <= 60000000000000.0)))) {
		tmp = -(y / (z / x));
	} else {
		tmp = (x / z) + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-2.5d+209)) .or. (.not. (y <= (-1.65d+154))) .and. (y <= (-1.19d+82)) .or. (.not. (y <= 60000000000000.0d0))) then
        tmp = -(y / (z / x))
    else
        tmp = (x / z) + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.5e+209) || (!(y <= -1.65e+154) && ((y <= -1.19e+82) || !(y <= 60000000000000.0)))) {
		tmp = -(y / (z / x));
	} else {
		tmp = (x / z) + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.5e+209) or (not (y <= -1.65e+154) and ((y <= -1.19e+82) or not (y <= 60000000000000.0))):
		tmp = -(y / (z / x))
	else:
		tmp = (x / z) + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.5e+209) || (!(y <= -1.65e+154) && ((y <= -1.19e+82) || !(y <= 60000000000000.0))))
		tmp = Float64(-Float64(y / Float64(z / x)));
	else
		tmp = Float64(Float64(x / z) + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.5e+209) || (~((y <= -1.65e+154)) && ((y <= -1.19e+82) || ~((y <= 60000000000000.0)))))
		tmp = -(y / (z / x));
	else
		tmp = (x / z) + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.5e+209], And[N[Not[LessEqual[y, -1.65e+154]], $MachinePrecision], Or[LessEqual[y, -1.19e+82], N[Not[LessEqual[y, 60000000000000.0]], $MachinePrecision]]]], (-N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), N[(N[(x / z), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.49999999999999982e209 or -1.65e154 < y < -1.1899999999999999e82 or 6e13 < y

   1. Initial program 74.6%

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

   2. Taylor expanded in x around inf 61.8%

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

      1. *-commutative 61.8%

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

      2. associate-/l* 66.2%

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

      3. mul-1-neg 66.2%

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

      4. unsub-neg 66.2%

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

   4. Simplified 66.2%

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

   5. Taylor expanded in y around inf 66.2%

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

      1. associate-*r/ 66.2%

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

      2. neg-mul-1 66.2%

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

   7. Simplified 66.2%

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

      1. frac-2neg 66.2%

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

      2. distribute-frac-neg 66.2%

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

      3. remove-double-neg 66.2%

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

      4. frac-2neg 66.2%

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

      5. distribute-frac-neg 66.2%

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

      6. frac-2neg 66.2%

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

      7. associate-/r/ 71.1%

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

      8. clear-num 71.0%

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

      9. associate-*l/ 71.1%

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

      10. *-un-lft-identity 71.1%

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

   9. Applied egg-rr 71.1%

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

   if -2.49999999999999982e209 < y < -1.65e154 or -1.1899999999999999e82 < y < 6e13

   1. Initial program 93.9%

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

   2. Taylor expanded in x around 0 98.5%

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

   3. Taylor expanded in y around 0 94.2%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+209} \lor \neg \left(y \leq -1.65 \cdot 10^{+154}\right) \land \left(y \leq -1.19 \cdot 10^{+82} \lor \neg \left(y \leq 60000000000000\right)\right):\\ \;\;\;\;-\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} + y\\ \end{array} \]

Alternative 3: 74.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+209}:\\ \;\;\;\;\frac{y}{z} \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{+153} \lor \neg \left(y \leq -1.19 \cdot 10^{+82}\right) \land y \leq 55000000000:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{else}:\\ \;\;\;\;-\frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.2e+209)
   (* (/ y z) (- x))
   (if (or (<= y -4.5e+153) (and (not (<= y -1.19e+82)) (<= y 55000000000.0)))
     (+ (/ x z) y)
     (- (/ y (/ z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.2e+209) {
		tmp = (y / z) * -x;
	} else if ((y <= -4.5e+153) || (!(y <= -1.19e+82) && (y <= 55000000000.0))) {
		tmp = (x / z) + y;
	} 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 (y <= (-4.2d+209)) then
        tmp = (y / z) * -x
    else if ((y <= (-4.5d+153)) .or. (.not. (y <= (-1.19d+82))) .and. (y <= 55000000000.0d0)) then
        tmp = (x / z) + y
    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 (y <= -4.2e+209) {
		tmp = (y / z) * -x;
	} else if ((y <= -4.5e+153) || (!(y <= -1.19e+82) && (y <= 55000000000.0))) {
		tmp = (x / z) + y;
	} else {
		tmp = -(y / (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.2e+209:
		tmp = (y / z) * -x
	elif (y <= -4.5e+153) or (not (y <= -1.19e+82) and (y <= 55000000000.0)):
		tmp = (x / z) + y
	else:
		tmp = -(y / (z / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.2e+209)
		tmp = Float64(Float64(y / z) * Float64(-x));
	elseif ((y <= -4.5e+153) || (!(y <= -1.19e+82) && (y <= 55000000000.0)))
		tmp = Float64(Float64(x / z) + y);
	else
		tmp = Float64(-Float64(y / Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.2e+209)
		tmp = (y / z) * -x;
	elseif ((y <= -4.5e+153) || (~((y <= -1.19e+82)) && (y <= 55000000000.0)))
		tmp = (x / z) + y;
	else
		tmp = -(y / (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.2e+209], N[(N[(y / z), $MachinePrecision] * (-x)), $MachinePrecision], If[Or[LessEqual[y, -4.5e+153], And[N[Not[LessEqual[y, -1.19e+82]], $MachinePrecision], LessEqual[y, 55000000000.0]]], N[(N[(x / z), $MachinePrecision] + y), $MachinePrecision], (-N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if y < -4.2e209

   1. Initial program 62.4%

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

   2. Taylor expanded in x around inf 52.1%

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

      1. *-commutative 52.1%

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

      2. associate-/l* 68.8%

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

      3. mul-1-neg 68.8%

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

      4. unsub-neg 68.8%

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

   4. Simplified 68.8%

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

   5. Taylor expanded in y around inf 68.8%

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

      1. associate-*r/ 68.8%

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

      2. neg-mul-1 68.8%

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

   7. Simplified 68.8%

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

      1. frac-2neg 68.8%

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

      2. div-inv 68.6%

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

      3. distribute-frac-neg 68.6%

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

      4. remove-double-neg 68.6%

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

      5. frac-2neg 68.6%

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

      6. clear-num 68.8%

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

      7. frac-2neg 68.8%

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

   9. Applied egg-rr 68.8%

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

   if -4.2e209 < y < -4.5000000000000001e153 or -1.1899999999999999e82 < y < 5.5e10

   1. Initial program 93.9%

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

   2. Taylor expanded in x around 0 98.5%

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

   3. Taylor expanded in y around 0 94.2%

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

   if -4.5000000000000001e153 < y < -1.1899999999999999e82 or 5.5e10 < y

   1. Initial program 78.0%

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

   2. Taylor expanded in x around inf 64.5%

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

      1. *-commutative 64.5%

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

      2. associate-/l* 65.5%

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

      3. mul-1-neg 65.5%

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

      4. unsub-neg 65.5%

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

   4. Simplified 65.5%

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

   5. Taylor expanded in y around inf 65.5%

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

      1. associate-*r/ 65.5%

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

      2. neg-mul-1 65.5%

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

   7. Simplified 65.5%

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

      1. frac-2neg 65.5%

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

      2. distribute-frac-neg 65.5%

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

      3. remove-double-neg 65.5%

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

      4. frac-2neg 65.5%

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

      5. distribute-frac-neg 65.5%

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

      6. frac-2neg 65.5%

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

      7. associate-/r/ 71.8%

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

      8. clear-num 71.7%

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

      9. associate-*l/ 71.8%

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

      10. *-un-lft-identity 71.8%

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

   9. Applied egg-rr 71.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+209}:\\ \;\;\;\;\frac{y}{z} \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{+153} \lor \neg \left(y \leq -1.19 \cdot 10^{+82}\right) \land y \leq 55000000000:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{else}:\\ \;\;\;\;-\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 4: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+27} \lor \neg \left(y \leq 90000000\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2e+27) (not (<= y 90000000.0)))
   (/ y (/ z (- z x)))
   (/ (+ x (* y (- z x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2e+27) || !(y <= 90000000.0)) {
		tmp = y / (z / (z - x));
	} else {
		tmp = (x + (y * (z - 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 <= (-2d+27)) .or. (.not. (y <= 90000000.0d0))) then
        tmp = y / (z / (z - x))
    else
        tmp = (x + (y * (z - x))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2e+27) || !(y <= 90000000.0)) {
		tmp = y / (z / (z - x));
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2e+27) or not (y <= 90000000.0):
		tmp = y / (z / (z - x))
	else:
		tmp = (x + (y * (z - x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2e+27) || !(y <= 90000000.0))
		tmp = Float64(y / Float64(z / Float64(z - x)));
	else
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2e+27) || ~((y <= 90000000.0)))
		tmp = y / (z / (z - x));
	else
		tmp = (x + (y * (z - x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2e+27], N[Not[LessEqual[y, 90000000.0]], $MachinePrecision]], N[(y / N[(z / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2e27 or 9e7 < y

   1. Initial program 73.4%

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

   2. Taylor expanded in y around inf 73.4%

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

      1. associate-/l* 99.9%

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

   4. Simplified 99.9%

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

   if -2e27 < y < 9e7

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+27} \lor \neg \left(y \leq 90000000\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]

Alternative 5: 94.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+17} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\left(z - x\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4e+17) (not (<= y 1.0))) (* (- z x) (/ y z)) (+ (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4e+17) || !(y <= 1.0)) {
		tmp = (z - x) * (y / z);
	} else {
		tmp = (x / z) + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-4d+17)) .or. (.not. (y <= 1.0d0))) then
        tmp = (z - x) * (y / z)
    else
        tmp = (x / z) + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4e+17) || !(y <= 1.0)) {
		tmp = (z - x) * (y / z);
	} else {
		tmp = (x / z) + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4e+17) or not (y <= 1.0):
		tmp = (z - x) * (y / z)
	else:
		tmp = (x / z) + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4e+17) || !(y <= 1.0))
		tmp = Float64(Float64(z - x) * Float64(y / z));
	else
		tmp = Float64(Float64(x / z) + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4e+17) || ~((y <= 1.0)))
		tmp = (z - x) * (y / z);
	else
		tmp = (x / z) + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4e+17], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(N[(z - x), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4e17 or 1 < y

   1. Initial program 73.6%

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

   2. Taylor expanded in y around inf 73.6%

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

      1. associate-/l* 99.9%

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

      2. associate-/r/ 92.7%

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

   4. Applied egg-rr 92.7%

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

   if -4e17 < y < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.7%

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

   3. Taylor expanded in y around 0 99.1%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+17} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\left(z - x\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} + y\\ \end{array} \]

Alternative 6: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+17} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4e+17) (not (<= y 1.0))) (/ y (/ z (- z x))) (+ (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4e+17) || !(y <= 1.0)) {
		tmp = y / (z / (z - x));
	} else {
		tmp = (x / z) + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-4d+17)) .or. (.not. (y <= 1.0d0))) then
        tmp = y / (z / (z - x))
    else
        tmp = (x / z) + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4e+17) || !(y <= 1.0)) {
		tmp = y / (z / (z - x));
	} else {
		tmp = (x / z) + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4e+17) or not (y <= 1.0):
		tmp = y / (z / (z - x))
	else:
		tmp = (x / z) + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4e+17) || !(y <= 1.0))
		tmp = Float64(y / Float64(z / Float64(z - x)));
	else
		tmp = Float64(Float64(x / z) + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4e+17) || ~((y <= 1.0)))
		tmp = y / (z / (z - x));
	else
		tmp = (x / z) + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4e+17], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y / N[(z / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4e17 or 1 < y

   1. Initial program 73.6%

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

   2. Taylor expanded in y around inf 73.6%

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

      1. associate-/l* 99.9%

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

   4. Simplified 99.9%

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

   if -4e17 < y < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.7%

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

   3. Taylor expanded in y around 0 99.1%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+17} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} + y\\ \end{array} \]

Alternative 7: 61.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-57}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.2e-57) y (if (<= y 2.6e-24) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.2e-57) {
		tmp = y;
	} else if (y <= 2.6e-24) {
		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 <= (-7.2d-57)) then
        tmp = y
    else if (y <= 2.6d-24) 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 <= -7.2e-57) {
		tmp = y;
	} else if (y <= 2.6e-24) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.2e-57:
		tmp = y
	elif y <= 2.6e-24:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.2e-57)
		tmp = y;
	elseif (y <= 2.6e-24)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7.2e-57)
		tmp = y;
	elseif (y <= 2.6e-24)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.2e-57], y, If[LessEqual[y, 2.6e-24], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -7.2000000000000005e-57 or 2.6e-24 < y

   1. Initial program 77.6%

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

   2. Taylor expanded in x around 0 43.5%

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

   if -7.2000000000000005e-57 < y < 2.6e-24

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 79.4%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-57}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 8: 77.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

2. Taylor expanded in x around 0 96.1%

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

3. Taylor expanded in y around 0 71.2%

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

4. Final simplification 71.2%

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

Alternative 9: 40.7% 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 86.4%

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

2. Taylor expanded in x around 0 35.2%

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

3. Final simplification 35.2%

   \[\leadsto y \]

Developer target: 93.9% 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 2023199 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

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