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

Percentage Accurate: 100.0% → 100.0%

Time: 6.2s

Alternatives: 9

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 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 98.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{-y}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))) (t_1 (/ x (- z y))))
   (if (<= t_0 -20.0)
     t_1
     (if (<= t_0 0.8) (/ (- x y) z) (if (<= t_0 2.0) (/ (- y) (- z y)) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = t_1;
	} else if (t_0 <= 0.8) {
		tmp = (x - y) / z;
	} else if (t_0 <= 2.0) {
		tmp = -y / (z - y);
	} else {
		tmp = t_1;
	}
	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 - y) / (z - y)
    t_1 = x / (z - y)
    if (t_0 <= (-20.0d0)) then
        tmp = t_1
    else if (t_0 <= 0.8d0) then
        tmp = (x - y) / z
    else if (t_0 <= 2.0d0) then
        tmp = -y / (z - y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = t_1;
	} else if (t_0 <= 0.8) {
		tmp = (x - y) / z;
	} else if (t_0 <= 2.0) {
		tmp = -y / (z - y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	t_1 = x / (z - y)
	tmp = 0
	if t_0 <= -20.0:
		tmp = t_1
	elif t_0 <= 0.8:
		tmp = (x - y) / z
	elif t_0 <= 2.0:
		tmp = -y / (z - y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	t_1 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (t_0 <= -20.0)
		tmp = t_1;
	elseif (t_0 <= 0.8)
		tmp = Float64(Float64(x - y) / z);
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64(-y) / Float64(z - y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	t_1 = x / (z - y);
	tmp = 0.0;
	if (t_0 <= -20.0)
		tmp = t_1;
	elseif (t_0 <= 0.8)
		tmp = (x - y) / z;
	elseif (t_0 <= 2.0)
		tmp = -y / (z - y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20.0], t$95$1, If[LessEqual[t$95$0, 0.8], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[((-y) / N[(z - y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -20 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 97.2

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

   5. Applied rewrites 97.2%

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

   if -20 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.80000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if 0.80000000000000004 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.8

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

   5. Applied rewrites 98.8%

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

Alternative 3: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{z}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))) (t_1 (/ x (- z y))))
   (if (<= t_0 -20.0)
     t_1
     (if (<= t_0 0.8) (/ (- x y) z) (if (<= t_0 2.0) (+ (/ z y) 1.0) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = t_1;
	} else if (t_0 <= 0.8) {
		tmp = (x - y) / z;
	} else if (t_0 <= 2.0) {
		tmp = (z / y) + 1.0;
	} else {
		tmp = t_1;
	}
	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 - y) / (z - y)
    t_1 = x / (z - y)
    if (t_0 <= (-20.0d0)) then
        tmp = t_1
    else if (t_0 <= 0.8d0) then
        tmp = (x - y) / z
    else if (t_0 <= 2.0d0) then
        tmp = (z / y) + 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = t_1;
	} else if (t_0 <= 0.8) {
		tmp = (x - y) / z;
	} else if (t_0 <= 2.0) {
		tmp = (z / y) + 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	t_1 = x / (z - y)
	tmp = 0
	if t_0 <= -20.0:
		tmp = t_1
	elif t_0 <= 0.8:
		tmp = (x - y) / z
	elif t_0 <= 2.0:
		tmp = (z / y) + 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	t_1 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (t_0 <= -20.0)
		tmp = t_1;
	elseif (t_0 <= 0.8)
		tmp = Float64(Float64(x - y) / z);
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64(z / y) + 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	t_1 = x / (z - y);
	tmp = 0.0;
	if (t_0 <= -20.0)
		tmp = t_1;
	elseif (t_0 <= 0.8)
		tmp = (x - y) / z;
	elseif (t_0 <= 2.0)
		tmp = (z / y) + 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20.0], t$95$1, If[LessEqual[t$95$0, 0.8], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(z / y), $MachinePrecision] + 1.0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -20 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 97.2

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

   5. Applied rewrites 97.2%

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

   if -20 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.80000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if 0.80000000000000004 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.2%

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

Alternative 4: 84.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_0 \leq 0.8 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))))
   (if (or (<= t_0 0.8) (not (<= t_0 2.0))) (/ x (- z y)) (+ (/ z y) 1.0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if ((t_0 <= 0.8) || !(t_0 <= 2.0)) {
		tmp = x / (z - y);
	} else {
		tmp = (z / y) + 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) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (z - y)
    if ((t_0 <= 0.8d0) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = x / (z - y)
    else
        tmp = (z / y) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if ((t_0 <= 0.8) || !(t_0 <= 2.0)) {
		tmp = x / (z - y);
	} else {
		tmp = (z / y) + 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	tmp = 0
	if (t_0 <= 0.8) or not (t_0 <= 2.0):
		tmp = x / (z - y)
	else:
		tmp = (z / y) + 1.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if ((t_0 <= 0.8) || !(t_0 <= 2.0))
		tmp = Float64(x / Float64(z - y));
	else
		tmp = Float64(Float64(z / y) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	tmp = 0.0;
	if ((t_0 <= 0.8) || ~((t_0 <= 2.0)))
		tmp = x / (z - y);
	else
		tmp = (z / y) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.8], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], N[(N[(z / y), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.80000000000000004 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 77.2

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

   5. Applied rewrites 77.2%

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

   if 0.80000000000000004 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.2%

      \[\leadsto \frac{z}{y} + \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.8 \lor \neg \left(\frac{x - y}{z - y} \leq 2\right):\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_0 \leq 0.8 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))))
   (if (or (<= t_0 0.8) (not (<= t_0 2.0))) (/ x z) (+ (/ z y) 1.0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if ((t_0 <= 0.8) || !(t_0 <= 2.0)) {
		tmp = x / z;
	} else {
		tmp = (z / y) + 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) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (z - y)
    if ((t_0 <= 0.8d0) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = x / z
    else
        tmp = (z / y) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if ((t_0 <= 0.8) || !(t_0 <= 2.0)) {
		tmp = x / z;
	} else {
		tmp = (z / y) + 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	tmp = 0
	if (t_0 <= 0.8) or not (t_0 <= 2.0):
		tmp = x / z
	else:
		tmp = (z / y) + 1.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if ((t_0 <= 0.8) || !(t_0 <= 2.0))
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(z / y) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	tmp = 0.0;
	if ((t_0 <= 0.8) || ~((t_0 <= 2.0)))
		tmp = x / z;
	else
		tmp = (z / y) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.8], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(x / z), $MachinePrecision], N[(N[(z / y), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.80000000000000004 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 60.2

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

   5. Applied rewrites 60.2%

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

   if 0.80000000000000004 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.2%

      \[\leadsto \frac{z}{y} + \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.8 \lor \neg \left(\frac{x - y}{z - y} \leq 2\right):\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_0 \leq 0.8 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))))
   (if (or (<= t_0 0.8) (not (<= t_0 2.0))) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if ((t_0 <= 0.8) || !(t_0 <= 2.0)) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (z - y)
    if ((t_0 <= 0.8d0) .or. (.not. (t_0 <= 2.0d0))) 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 t_0 = (x - y) / (z - y);
	double tmp;
	if ((t_0 <= 0.8) || !(t_0 <= 2.0)) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	tmp = 0
	if (t_0 <= 0.8) or not (t_0 <= 2.0):
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if ((t_0 <= 0.8) || !(t_0 <= 2.0))
		tmp = Float64(x / z);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	tmp = 0.0;
	if ((t_0 <= 0.8) || ~((t_0 <= 2.0)))
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.8], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.80000000000000004 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 60.2

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

   5. Applied rewrites 60.2%

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

   if 0.80000000000000004 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 95.3%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.8 \lor \neg \left(\frac{x - y}{z - y} \leq 2\right):\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+46} \lor \neg \left(y \leq 1.3 \cdot 10^{-53}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.8e+46) (not (<= y 1.3e-53))) (- 1.0 (/ x y)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.8e+46) || !(y <= 1.3e-53)) {
		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 <= (-6.8d+46)) .or. (.not. (y <= 1.3d-53))) 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 <= -6.8e+46) || !(y <= 1.3e-53)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6.8e+46) or not (y <= 1.3e-53):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6.8e+46) || !(y <= 1.3e-53))
		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 <= -6.8e+46) || ~((y <= 1.3e-53)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6.8e+46], N[Not[LessEqual[y, 1.3e-53]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.7999999999999996e46 or 1.29999999999999998e-53 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. div-add N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-inverses N/A

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

      9. +-commutative N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 73.8

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

   8. Applied rewrites 73.8%

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

   if -6.7999999999999996e46 < y < 1.29999999999999998e-53

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 77.2

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

   5. Applied rewrites 77.2%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+46} \lor \neg \left(y \leq 1.3 \cdot 10^{-53}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 99.9%

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

Alternative 9: 34.8% accurate, 18.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 35.7%

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

Developer Target 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]

Reproduce

herbie shell --seed 2024337 
(FPCore (x y z)
  :name "Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ x (- z y)) (/ y (- z y))))

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