Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{z - y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-sub 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{x}{z - y} - \frac{y}{z - y}} \]
5. Add Preprocessing

Alternative 2: 75.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{z - y}\\ t_1 := \frac{y}{y - z}\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{+45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (- z y))) (t_1 (/ y (- y z))))
   (if (<= x -9.5e+45)
     t_0
     (if (<= x 2.8e-127)
       t_1
       (if (<= x 8.4e-88) (/ x z) (if (<= x 9.2e+34) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x / (z - y);
	double t_1 = y / (y - z);
	double tmp;
	if (x <= -9.5e+45) {
		tmp = t_0;
	} else if (x <= 2.8e-127) {
		tmp = t_1;
	} else if (x <= 8.4e-88) {
		tmp = x / z;
	} else if (x <= 9.2e+34) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (z - y)
    t_1 = y / (y - z)
    if (x <= (-9.5d+45)) then
        tmp = t_0
    else if (x <= 2.8d-127) then
        tmp = t_1
    else if (x <= 8.4d-88) then
        tmp = x / z
    else if (x <= 9.2d+34) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x / (z - y);
	double t_1 = y / (y - z);
	double tmp;
	if (x <= -9.5e+45) {
		tmp = t_0;
	} else if (x <= 2.8e-127) {
		tmp = t_1;
	} else if (x <= 8.4e-88) {
		tmp = x / z;
	} else if (x <= 9.2e+34) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / (z - y)
	t_1 = y / (y - z)
	tmp = 0
	if x <= -9.5e+45:
		tmp = t_0
	elif x <= 2.8e-127:
		tmp = t_1
	elif x <= 8.4e-88:
		tmp = x / z
	elif x <= 9.2e+34:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x / Float64(z - y))
	t_1 = Float64(y / Float64(y - z))
	tmp = 0.0
	if (x <= -9.5e+45)
		tmp = t_0;
	elseif (x <= 2.8e-127)
		tmp = t_1;
	elseif (x <= 8.4e-88)
		tmp = Float64(x / z);
	elseif (x <= 9.2e+34)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x / (z - y);
	t_1 = y / (y - z);
	tmp = 0.0;
	if (x <= -9.5e+45)
		tmp = t_0;
	elseif (x <= 2.8e-127)
		tmp = t_1;
	elseif (x <= 8.4e-88)
		tmp = x / z;
	elseif (x <= 9.2e+34)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.5e+45], t$95$0, If[LessEqual[x, 2.8e-127], t$95$1, If[LessEqual[x, 8.4e-88], N[(x / z), $MachinePrecision], If[LessEqual[x, 9.2e+34], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.4999999999999998e45 or 9.1999999999999993e34 < x

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 85.4%

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

   if -9.4999999999999998e45 < x < 2.8e-127 or 8.3999999999999998e-88 < x < 9.1999999999999993e34

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 81.9%

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

      1. neg-mul-1 81.9%

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

      2. distribute-neg-frac 81.9%

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

   5. Simplified 81.9%

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

      1. frac-2neg 81.9%

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

      2. div-inv 81.7%

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

      3. remove-double-neg 81.7%

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

      4. sub-neg 81.7%

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

      5. distribute-neg-in 81.7%

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

      6. remove-double-neg 81.7%

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

   7. Applied egg-rr 81.7%

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

      1. associate-*r/ 81.9%

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

      2. *-rgt-identity 81.9%

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

      3. +-commutative 81.9%

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

      4. sub-neg 81.9%

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

   9. Simplified 81.9%

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

   if 2.8e-127 < x < 8.3999999999999998e-88

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 100.0%

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

Alternative 3: 76.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+36} \lor \neg \left(y \leq 8.5 \cdot 10^{-23}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.4e+36) (not (<= y 8.5e-23))) (- 1.0 (/ x y)) (/ x (- z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.4e+36) || !(y <= 8.5e-23)) {
		tmp = 1.0 - (x / y);
	} 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 <= (-2.4d+36)) .or. (.not. (y <= 8.5d-23))) then
        tmp = 1.0d0 - (x / y)
    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 <= -2.4e+36) || !(y <= 8.5e-23)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (z - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.4e+36) or not (y <= 8.5e-23):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / (z - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.4e+36) || !(y <= 8.5e-23))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / Float64(z - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.4e+36) || ~((y <= 8.5e-23)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.4e+36], N[Not[LessEqual[y, 8.5e-23]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.39999999999999992e36 or 8.4999999999999996e-23 < y

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 72.1%

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

      1. div-sub 72.1%

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

      2. sub-neg 72.1%

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

      3. *-inverses 72.1%

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

      4. metadata-eval 72.1%

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

      5. distribute-lft-in 72.1%

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

      6. metadata-eval 72.1%

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

      7. +-commutative 72.1%

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

      8. mul-1-neg 72.1%

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

      9. unsub-neg 72.1%

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

   5. Simplified 72.1%

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

   if -2.39999999999999992e36 < y < 8.4999999999999996e-23

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 79.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+36} \lor \neg \left(y \leq 8.5 \cdot 10^{-23}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-56} \lor \neg \left(y \leq 4.8 \cdot 10^{-109}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5e-56) (not (<= y 4.8e-109))) (- 1.0 (/ x y)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5e-56) || !(y <= 4.8e-109)) {
		tmp = 1.0 - (x / y);
	} 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 <= (-5d-56)) .or. (.not. (y <= 4.8d-109))) then
        tmp = 1.0d0 - (x / y)
    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 <= -5e-56) || !(y <= 4.8e-109)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5e-56) or not (y <= 4.8e-109):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5e-56) || !(y <= 4.8e-109))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5e-56) || ~((y <= 4.8e-109)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5e-56], N[Not[LessEqual[y, 4.8e-109]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999997e-56 or 4.79999999999999977e-109 < y

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 66.9%

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

      1. div-sub 67.0%

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

      2. sub-neg 67.0%

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

      3. *-inverses 67.0%

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

      4. metadata-eval 67.0%

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

      5. distribute-lft-in 67.0%

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

      6. metadata-eval 67.0%

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

      7. +-commutative 67.0%

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

      8. mul-1-neg 67.0%

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

      9. unsub-neg 67.0%

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

   5. Simplified 67.0%

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

   if -4.99999999999999997e-56 < y < 4.79999999999999977e-109

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 77.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-56} \lor \neg \left(y \leq 4.8 \cdot 10^{-109}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+39}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.6e+39) 1.0 (if (<= y 5.8e-24) (/ x z) (+ 1.0 (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.6e+39) {
		tmp = 1.0;
	} else if (y <= 5.8e-24) {
		tmp = x / z;
	} else {
		tmp = 1.0 + (z / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.6d+39)) then
        tmp = 1.0d0
    else if (y <= 5.8d-24) then
        tmp = x / z
    else
        tmp = 1.0d0 + (z / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.6e+39) {
		tmp = 1.0;
	} else if (y <= 5.8e-24) {
		tmp = x / z;
	} else {
		tmp = 1.0 + (z / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.6e+39:
		tmp = 1.0
	elif y <= 5.8e-24:
		tmp = x / z
	else:
		tmp = 1.0 + (z / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.6e+39)
		tmp = 1.0;
	elseif (y <= 5.8e-24)
		tmp = Float64(x / z);
	else
		tmp = Float64(1.0 + Float64(z / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.6e+39)
		tmp = 1.0;
	elseif (y <= 5.8e-24)
		tmp = x / z;
	else
		tmp = 1.0 + (z / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.6e+39], 1.0, If[LessEqual[y, 5.8e-24], N[(x / z), $MachinePrecision], N[(1.0 + N[(z / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.59999999999999996e39

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 61.3%

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

   if -1.59999999999999996e39 < y < 5.7999999999999997e-24

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 64.2%

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

   if 5.7999999999999997e-24 < y

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 69.9%

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

      1. associate-+r- 69.9%

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

      2. distribute-lft-out-- 69.9%

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

      3. div-sub 69.9%

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

      4. mul-1-neg 69.9%

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

      5. unsub-neg 69.9%

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

   5. Simplified 69.9%

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

   6. Taylor expanded in x around 0 49.1%

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

Alternative 6: 61.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+37}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.2e+37) 1.0 (if (<= y 2.8e-28) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.2e+37) {
		tmp = 1.0;
	} else if (y <= 2.8e-28) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5.2d+37)) then
        tmp = 1.0d0
    else if (y <= 2.8d-28) then
        tmp = x / z
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.2e+37) {
		tmp = 1.0;
	} else if (y <= 2.8e-28) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5.2e+37:
		tmp = 1.0
	elif y <= 2.8e-28:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.2e+37)
		tmp = 1.0;
	elseif (y <= 2.8e-28)
		tmp = Float64(x / z);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5.2e+37)
		tmp = 1.0;
	elseif (y <= 2.8e-28)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.2e+37], 1.0, If[LessEqual[y, 2.8e-28], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -5.1999999999999998e37 or 2.7999999999999998e-28 < y

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 54.0%

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

   if -5.1999999999999998e37 < y < 2.7999999999999998e-28

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 64.2%

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

Alternative 7: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{z - y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 8: 35.5% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{z - y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf 33.7%

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

Developer target: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024085 
(FPCore (x y z)
  :name "Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1"
  :precision binary64

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

  (/ (- x y) (- z y)))