Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A

Percentage Accurate: 99.7% → 100.0%

Time: 7.8s

Alternatives: 10

Speedup: 1.5×

• 20.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - 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 = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - 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 = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{\left(y + x\right) - z}{y} \cdot 4 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ (- (+ y x) z) y) 4.0))

C

double code(double x, double y, double z) {
	return (((y + x) - z) / y) * 4.0;
}

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 + x) - z) / y) * 4.0d0
end function

Java

public static double code(double x, double y, double z) {
	return (((y + x) - z) / y) * 4.0;
}

Python

def code(x, y, z):
	return (((y + x) - z) / y) * 4.0

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(y + x) - z) / y) * 4.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((y + x) - z) / y) * 4.0;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(y + x), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision] * 4.0), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. distribute-lft-out N/A

      \[\leadsto \frac{\color{blue}{4 \cdot \left(y + \left(x - z\right)\right)}}{y} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{4 \cdot \frac{y + \left(x - z\right)}{y}} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\frac{y + \left(x - z\right)}{y} \cdot 4} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{y + \left(x - z\right)}{y} \cdot 4} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y + \left(x - z\right)}{y}} \cdot 4 \]

   6. associate-+r- N/A

      \[\leadsto \frac{\color{blue}{\left(y + x\right) - z}}{y} \cdot 4 \]

   7. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(y + x\right) - z}}{y} \cdot 4 \]

   8. lower-+.f64 99.9

      \[\leadsto \frac{\color{blue}{\left(y + x\right)} - z}{y} \cdot 4 \]

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{\left(y + x\right) - z}{y} \cdot 4} \]
6. Add Preprocessing

Alternative 2: 66.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ t_1 := \frac{-4 \cdot z}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+202}:\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{elif}\;t\_0 \leq -500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 500000:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (- (+ (* 0.75 y) x) z) 4.0) y)) (t_1 (/ (* -4.0 z) y)))
   (if (<= t_0 -1e+202)
     (* (/ x y) 4.0)
     (if (<= t_0 -500000.0) t_1 (if (<= t_0 500000.0) 4.0 t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double t_1 = (-4.0 * z) / y;
	double tmp;
	if (t_0 <= -1e+202) {
		tmp = (x / y) * 4.0;
	} else if (t_0 <= -500000.0) {
		tmp = t_1;
	} else if (t_0 <= 500000.0) {
		tmp = 4.0;
	} 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 = ((((0.75d0 * y) + x) - z) * 4.0d0) / y
    t_1 = ((-4.0d0) * z) / y
    if (t_0 <= (-1d+202)) then
        tmp = (x / y) * 4.0d0
    else if (t_0 <= (-500000.0d0)) then
        tmp = t_1
    else if (t_0 <= 500000.0d0) then
        tmp = 4.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double t_1 = (-4.0 * z) / y;
	double tmp;
	if (t_0 <= -1e+202) {
		tmp = (x / y) * 4.0;
	} else if (t_0 <= -500000.0) {
		tmp = t_1;
	} else if (t_0 <= 500000.0) {
		tmp = 4.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((((0.75 * y) + x) - z) * 4.0) / y
	t_1 = (-4.0 * z) / y
	tmp = 0
	if t_0 <= -1e+202:
		tmp = (x / y) * 4.0
	elif t_0 <= -500000.0:
		tmp = t_1
	elif t_0 <= 500000.0:
		tmp = 4.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(0.75 * y) + x) - z) * 4.0) / y)
	t_1 = Float64(Float64(-4.0 * z) / y)
	tmp = 0.0
	if (t_0 <= -1e+202)
		tmp = Float64(Float64(x / y) * 4.0);
	elseif (t_0 <= -500000.0)
		tmp = t_1;
	elseif (t_0 <= 500000.0)
		tmp = 4.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((((0.75 * y) + x) - z) * 4.0) / y;
	t_1 = (-4.0 * z) / y;
	tmp = 0.0;
	if (t_0 <= -1e+202)
		tmp = (x / y) * 4.0;
	elseif (t_0 <= -500000.0)
		tmp = t_1;
	elseif (t_0 <= 500000.0)
		tmp = 4.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(0.75 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * z), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+202], N[(N[(x / y), $MachinePrecision] * 4.0), $MachinePrecision], If[LessEqual[t$95$0, -500000.0], t$95$1, If[LessEqual[t$95$0, 500000.0], 4.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -9.999999999999999e201

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out N/A

         \[\leadsto \frac{\color{blue}{4 \cdot \left(y + \left(x - z\right)\right)}}{y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{4 \cdot \frac{y + \left(x - z\right)}{y}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y + \left(x - z\right)}{y} \cdot 4} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y + \left(x - z\right)}{y} \cdot 4} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y + \left(x - z\right)}{y}} \cdot 4 \]

      6. associate-+r- N/A

         \[\leadsto \frac{\color{blue}{\left(y + x\right) - z}}{y} \cdot 4 \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y + x\right) - z}}{y} \cdot 4 \]

      8. lower-+.f64 100.0

         \[\leadsto \frac{\color{blue}{\left(y + x\right)} - z}{y} \cdot 4 \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\left(y + x\right) - z}{y} \cdot 4} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.6%

      \[\leadsto \frac{x}{y} \cdot 4 \]

   if -9.999999999999999e201 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -5e5 or 5e5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 99.2%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \frac{\frac{3}{4} \cdot y - z}{y} + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot y - z}{y} \cdot 4} + 1 \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{3}{4} \cdot y}{y} - \frac{z}{y}\right)} \cdot 4 + 1 \]

      4. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\frac{3}{4} \cdot \frac{y}{y}} - \frac{z}{y}\right) \cdot 4 + 1 \]

      5. *-inverses N/A

         \[\leadsto \left(\frac{3}{4} \cdot \color{blue}{1} - \frac{z}{y}\right) \cdot 4 + 1 \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{3}{4}} - \frac{z}{y}\right) \cdot 4 + 1 \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(\frac{3}{4} - \frac{z}{y}\right)} + 1 \]

      8. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{3}{4} + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)\right)} + 1 \]

      9. +-commutative N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) + \frac{3}{4}\right)} + 1 \]

      10. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4 + \frac{3}{4} \cdot 4\right)} + 1 \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{4 \cdot \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      12. *-lft-identity N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{1 \cdot z}}{y}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      13. associate-*l/ N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{y} \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(\frac{1}{y} \cdot z\right)\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      15. associate-*l* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot \frac{1}{y}\right) \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      16. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot \left(4 \cdot \frac{1}{y}\right)}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      17. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(z \cdot \left(4 \cdot \frac{1}{y}\right)\right)\right) + \color{blue}{3}\right) + 1 \]

      18. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z \cdot \left(4 \cdot \frac{1}{y}\right)\right)\right) + \left(3 + 1\right)} \]

      19. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(z \cdot \left(4 \cdot \frac{1}{y}\right)\right)\right) + \color{blue}{4} \]

   5. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 55.6%

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

   10. Applied rewrites 55.8%

       \[\leadsto \frac{-4 \cdot z}{y} \]

   if -5e5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5e5

   1. Initial program 99.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 95.3%

      \[\leadsto \color{blue}{4} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -1 \cdot 10^{+202}:\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -500000:\\ \;\;\;\;\frac{-4 \cdot z}{y}\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 500000:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - z}{y} \cdot 4\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -4000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (- x z) y) 4.0)) (t_1 (/ (* (- (+ (* 0.75 y) x) z) 4.0) y)))
   (if (<= t_1 -4000000.0)
     t_0
     (if (<= t_1 500000.0) (fma 4.0 (/ x y) 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((x - z) / y) * 4.0;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -4000000.0) {
		tmp = t_0;
	} else if (t_1 <= 500000.0) {
		tmp = fma(4.0, (x / y), 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x - z) / y) * 4.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(0.75 * y) + x) - z) * 4.0) / y)
	tmp = 0.0
	if (t_1 <= -4000000.0)
		tmp = t_0;
	elseif (t_1 <= 500000.0)
		tmp = fma(4.0, Float64(x / y), 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(0.75 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -4000000.0], t$95$0, If[LessEqual[t$95$1, 500000.0], N[(4.0 * N[(x / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -4e6 or 5e5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 99.4%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out N/A

         \[\leadsto \frac{\color{blue}{4 \cdot \left(y + \left(x - z\right)\right)}}{y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{4 \cdot \frac{y + \left(x - z\right)}{y}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y + \left(x - z\right)}{y} \cdot 4} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y + \left(x - z\right)}{y} \cdot 4} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y + \left(x - z\right)}{y}} \cdot 4 \]

      6. associate-+r- N/A

         \[\leadsto \frac{\color{blue}{\left(y + x\right) - z}}{y} \cdot 4 \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y + x\right) - z}}{y} \cdot 4 \]

      8. lower-+.f64 100.0

         \[\leadsto \frac{\color{blue}{\left(y + x\right)} - z}{y} \cdot 4 \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\left(y + x\right) - z}{y} \cdot 4} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.0%

      \[\leadsto \frac{x - z}{y} \cdot 4 \]

   if -4e6 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5e5

   1. Initial program 99.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \frac{3}{4} \cdot y}{y}} \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{-4}{y}, x, -4\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.1%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -4000000:\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 500000:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - z\right) \cdot \frac{4}{y}\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -4000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- x z) (/ 4.0 y))) (t_1 (/ (* (- (+ (* 0.75 y) x) z) 4.0) y)))
   (if (<= t_1 -4000000.0)
     t_0
     (if (<= t_1 500000.0) (fma 4.0 (/ x y) 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - z) * (4.0 / y);
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -4000000.0) {
		tmp = t_0;
	} else if (t_1 <= 500000.0) {
		tmp = fma(4.0, (x / y), 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - z) * Float64(4.0 / y))
	t_1 = Float64(Float64(Float64(Float64(Float64(0.75 * y) + x) - z) * 4.0) / y)
	tmp = 0.0
	if (t_1 <= -4000000.0)
		tmp = t_0;
	elseif (t_1 <= 500000.0)
		tmp = fma(4.0, Float64(x / y), 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(0.75 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -4000000.0], t$95$0, If[LessEqual[t$95$1, 500000.0], N[(4.0 * N[(x / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -4e6 or 5e5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 99.4%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out N/A

         \[\leadsto \frac{\color{blue}{4 \cdot \left(y + \left(x - z\right)\right)}}{y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{4 \cdot \frac{y + \left(x - z\right)}{y}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y + \left(x - z\right)}{y} \cdot 4} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y + \left(x - z\right)}{y} \cdot 4} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y + \left(x - z\right)}{y}} \cdot 4 \]

      6. associate-+r- N/A

         \[\leadsto \frac{\color{blue}{\left(y + x\right) - z}}{y} \cdot 4 \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y + x\right) - z}}{y} \cdot 4 \]

      8. lower-+.f64 100.0

         \[\leadsto \frac{\color{blue}{\left(y + x\right)} - z}{y} \cdot 4 \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\left(y + x\right) - z}{y} \cdot 4} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.0%

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

   10. Applied rewrites 97.8%

       \[\leadsto \color{blue}{\left(x - z\right) \cdot \frac{4}{y}} \]

   if -4e6 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5e5

   1. Initial program 99.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \frac{3}{4} \cdot y}{y}} \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{-4}{y}, x, -4\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -4000000:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 500000:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4 \cdot z}{y}\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -500000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* -4.0 z) y)) (t_1 (/ (* (- (+ (* 0.75 y) x) z) 4.0) y)))
   (if (<= t_1 -500000.0) t_0 (if (<= t_1 500000.0) 4.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-4.0 * z) / y;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -500000.0) {
		tmp = t_0;
	} else if (t_1 <= 500000.0) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-4.0d0) * z) / y
    t_1 = ((((0.75d0 * y) + x) - z) * 4.0d0) / y
    if (t_1 <= (-500000.0d0)) then
        tmp = t_0
    else if (t_1 <= 500000.0d0) then
        tmp = 4.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-4.0 * z) / y;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -500000.0) {
		tmp = t_0;
	} else if (t_1 <= 500000.0) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-4.0 * z) / y
	t_1 = ((((0.75 * y) + x) - z) * 4.0) / y
	tmp = 0
	if t_1 <= -500000.0:
		tmp = t_0
	elif t_1 <= 500000.0:
		tmp = 4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-4.0 * z) / y)
	t_1 = Float64(Float64(Float64(Float64(Float64(0.75 * y) + x) - z) * 4.0) / y)
	tmp = 0.0
	if (t_1 <= -500000.0)
		tmp = t_0;
	elseif (t_1 <= 500000.0)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-4.0 * z) / y;
	t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	tmp = 0.0;
	if (t_1 <= -500000.0)
		tmp = t_0;
	elseif (t_1 <= 500000.0)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-4.0 * z), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(0.75 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -500000.0], t$95$0, If[LessEqual[t$95$1, 500000.0], 4.0, t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -5e5 or 5e5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 99.4%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \frac{\frac{3}{4} \cdot y - z}{y} + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot y - z}{y} \cdot 4} + 1 \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{3}{4} \cdot y}{y} - \frac{z}{y}\right)} \cdot 4 + 1 \]

      4. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\frac{3}{4} \cdot \frac{y}{y}} - \frac{z}{y}\right) \cdot 4 + 1 \]

      5. *-inverses N/A

         \[\leadsto \left(\frac{3}{4} \cdot \color{blue}{1} - \frac{z}{y}\right) \cdot 4 + 1 \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{3}{4}} - \frac{z}{y}\right) \cdot 4 + 1 \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(\frac{3}{4} - \frac{z}{y}\right)} + 1 \]

      8. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{3}{4} + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)\right)} + 1 \]

      9. +-commutative N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) + \frac{3}{4}\right)} + 1 \]

      10. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4 + \frac{3}{4} \cdot 4\right)} + 1 \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{4 \cdot \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      12. *-lft-identity N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{1 \cdot z}}{y}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      13. associate-*l/ N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{y} \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(\frac{1}{y} \cdot z\right)\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      15. associate-*l* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot \frac{1}{y}\right) \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      16. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot \left(4 \cdot \frac{1}{y}\right)}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      17. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(z \cdot \left(4 \cdot \frac{1}{y}\right)\right)\right) + \color{blue}{3}\right) + 1 \]

      18. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z \cdot \left(4 \cdot \frac{1}{y}\right)\right)\right) + \left(3 + 1\right)} \]

      19. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(z \cdot \left(4 \cdot \frac{1}{y}\right)\right)\right) + \color{blue}{4} \]

   5. Applied rewrites 55.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 53.0%

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

   10. Applied rewrites 53.1%

       \[\leadsto \frac{-4 \cdot z}{y} \]

   if -5e5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5e5

   1. Initial program 99.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 95.3%

      \[\leadsto \color{blue}{4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -500000:\\ \;\;\;\;\frac{-4 \cdot z}{y}\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 500000:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4}{y} \cdot z\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -500000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ -4.0 y) z)) (t_1 (/ (* (- (+ (* 0.75 y) x) z) 4.0) y)))
   (if (<= t_1 -500000.0) t_0 (if (<= t_1 500000.0) 4.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-4.0 / y) * z;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -500000.0) {
		tmp = t_0;
	} else if (t_1 <= 500000.0) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-4.0d0) / y) * z
    t_1 = ((((0.75d0 * y) + x) - z) * 4.0d0) / y
    if (t_1 <= (-500000.0d0)) then
        tmp = t_0
    else if (t_1 <= 500000.0d0) then
        tmp = 4.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-4.0 / y) * z;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -500000.0) {
		tmp = t_0;
	} else if (t_1 <= 500000.0) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-4.0 / y) * z
	t_1 = ((((0.75 * y) + x) - z) * 4.0) / y
	tmp = 0
	if t_1 <= -500000.0:
		tmp = t_0
	elif t_1 <= 500000.0:
		tmp = 4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-4.0 / y) * z)
	t_1 = Float64(Float64(Float64(Float64(Float64(0.75 * y) + x) - z) * 4.0) / y)
	tmp = 0.0
	if (t_1 <= -500000.0)
		tmp = t_0;
	elseif (t_1 <= 500000.0)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-4.0 / y) * z;
	t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	tmp = 0.0;
	if (t_1 <= -500000.0)
		tmp = t_0;
	elseif (t_1 <= 500000.0)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-4.0 / y), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(0.75 * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision] * 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -500000.0], t$95$0, If[LessEqual[t$95$1, 500000.0], 4.0, t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -5e5 or 5e5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 99.4%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \frac{\frac{3}{4} \cdot y - z}{y} + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot y - z}{y} \cdot 4} + 1 \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{3}{4} \cdot y}{y} - \frac{z}{y}\right)} \cdot 4 + 1 \]

      4. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\frac{3}{4} \cdot \frac{y}{y}} - \frac{z}{y}\right) \cdot 4 + 1 \]

      5. *-inverses N/A

         \[\leadsto \left(\frac{3}{4} \cdot \color{blue}{1} - \frac{z}{y}\right) \cdot 4 + 1 \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{3}{4}} - \frac{z}{y}\right) \cdot 4 + 1 \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(\frac{3}{4} - \frac{z}{y}\right)} + 1 \]

      8. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{3}{4} + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)\right)} + 1 \]

      9. +-commutative N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) + \frac{3}{4}\right)} + 1 \]

      10. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4 + \frac{3}{4} \cdot 4\right)} + 1 \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{4 \cdot \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      12. *-lft-identity N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{1 \cdot z}}{y}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      13. associate-*l/ N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{y} \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(\frac{1}{y} \cdot z\right)\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      15. associate-*l* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot \frac{1}{y}\right) \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      16. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot \left(4 \cdot \frac{1}{y}\right)}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      17. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(z \cdot \left(4 \cdot \frac{1}{y}\right)\right)\right) + \color{blue}{3}\right) + 1 \]

      18. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z \cdot \left(4 \cdot \frac{1}{y}\right)\right)\right) + \left(3 + 1\right)} \]

      19. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(z \cdot \left(4 \cdot \frac{1}{y}\right)\right)\right) + \color{blue}{4} \]

   5. Applied rewrites 55.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 53.0%

      \[\leadsto \frac{-4}{y} \cdot \color{blue}{z} \]

   if -5e5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5e5

   1. Initial program 99.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 95.3%

      \[\leadsto \color{blue}{4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -500000:\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 500000:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(4, \frac{x}{y}, 4\right)\\ \mathbf{if}\;x \leq -8 \cdot 10^{+65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma 4.0 (/ x y) 4.0)))
   (if (<= x -8e+65) t_0 (if (<= x 1.8e+61) (fma -4.0 (/ z y) 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(4.0, (x / y), 4.0);
	double tmp;
	if (x <= -8e+65) {
		tmp = t_0;
	} else if (x <= 1.8e+61) {
		tmp = fma(-4.0, (z / y), 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(4.0, Float64(x / y), 4.0)
	tmp = 0.0
	if (x <= -8e+65)
		tmp = t_0;
	elseif (x <= 1.8e+61)
		tmp = fma(-4.0, Float64(z / y), 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(x / y), $MachinePrecision] + 4.0), $MachinePrecision]}, If[LessEqual[x, -8e+65], t$95$0, If[LessEqual[x, 1.8e+61], N[(-4.0 * N[(z / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.9999999999999999e65 or 1.80000000000000005e61 < x

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \frac{3}{4} \cdot y}{y}} \]

   5. Applied rewrites 88.0%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{-4}{y}, x, -4\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 88.2%

      \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{x}{y}}, 4\right) \]

   if -7.9999999999999999e65 < x < 1.80000000000000005e61

   1. Initial program 99.3%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \frac{\frac{3}{4} \cdot y - z}{y} + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot y - z}{y} \cdot 4} + 1 \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{3}{4} \cdot y}{y} - \frac{z}{y}\right)} \cdot 4 + 1 \]

      4. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\frac{3}{4} \cdot \frac{y}{y}} - \frac{z}{y}\right) \cdot 4 + 1 \]

      5. *-inverses N/A

         \[\leadsto \left(\frac{3}{4} \cdot \color{blue}{1} - \frac{z}{y}\right) \cdot 4 + 1 \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{3}{4}} - \frac{z}{y}\right) \cdot 4 + 1 \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(\frac{3}{4} - \frac{z}{y}\right)} + 1 \]

      8. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{3}{4} + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)\right)} + 1 \]

      9. +-commutative N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) + \frac{3}{4}\right)} + 1 \]

      10. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4 + \frac{3}{4} \cdot 4\right)} + 1 \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{4 \cdot \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      12. *-lft-identity N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{1 \cdot z}}{y}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      13. associate-*l/ N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{y} \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(\frac{1}{y} \cdot z\right)\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      15. associate-*l* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot \frac{1}{y}\right) \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      16. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot \left(4 \cdot \frac{1}{y}\right)}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      17. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(z \cdot \left(4 \cdot \frac{1}{y}\right)\right)\right) + \color{blue}{3}\right) + 1 \]

      18. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z \cdot \left(4 \cdot \frac{1}{y}\right)\right)\right) + \left(3 + 1\right)} \]

      19. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(z \cdot \left(4 \cdot \frac{1}{y}\right)\right)\right) + \color{blue}{4} \]

   5. Applied rewrites 89.1%

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

Alternative 8: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} \cdot 4\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{+163}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ x y) 4.0)))
   (if (<= x -8.2e+163) t_0 (if (<= x 1.5e+67) (fma -4.0 (/ z y) 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x / y) * 4.0;
	double tmp;
	if (x <= -8.2e+163) {
		tmp = t_0;
	} else if (x <= 1.5e+67) {
		tmp = fma(-4.0, (z / y), 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / y) * 4.0)
	tmp = 0.0
	if (x <= -8.2e+163)
		tmp = t_0;
	elseif (x <= 1.5e+67)
		tmp = fma(-4.0, Float64(z / y), 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[x, -8.2e+163], t$95$0, If[LessEqual[x, 1.5e+67], N[(-4.0 * N[(z / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.1999999999999998e163 or 1.50000000000000005e67 < x

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-out N/A

         \[\leadsto \frac{\color{blue}{4 \cdot \left(y + \left(x - z\right)\right)}}{y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{4 \cdot \frac{y + \left(x - z\right)}{y}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y + \left(x - z\right)}{y} \cdot 4} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y + \left(x - z\right)}{y} \cdot 4} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y + \left(x - z\right)}{y}} \cdot 4 \]

      6. associate-+r- N/A

         \[\leadsto \frac{\color{blue}{\left(y + x\right) - z}}{y} \cdot 4 \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y + x\right) - z}}{y} \cdot 4 \]

      8. lower-+.f64 99.9

         \[\leadsto \frac{\color{blue}{\left(y + x\right)} - z}{y} \cdot 4 \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\left(y + x\right) - z}{y} \cdot 4} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 76.8%

      \[\leadsto \frac{x}{y} \cdot 4 \]

   if -8.1999999999999998e163 < x < 1.50000000000000005e67

   1. Initial program 99.3%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \frac{\frac{3}{4} \cdot y - z}{y} + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{3}{4} \cdot y - z}{y} \cdot 4} + 1 \]

      3. div-sub N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{3}{4} \cdot y}{y} - \frac{z}{y}\right)} \cdot 4 + 1 \]

      4. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\frac{3}{4} \cdot \frac{y}{y}} - \frac{z}{y}\right) \cdot 4 + 1 \]

      5. *-inverses N/A

         \[\leadsto \left(\frac{3}{4} \cdot \color{blue}{1} - \frac{z}{y}\right) \cdot 4 + 1 \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{3}{4}} - \frac{z}{y}\right) \cdot 4 + 1 \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(\frac{3}{4} - \frac{z}{y}\right)} + 1 \]

      8. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{3}{4} + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)\right)} + 1 \]

      9. +-commutative N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) + \frac{3}{4}\right)} + 1 \]

      10. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4 + \frac{3}{4} \cdot 4\right)} + 1 \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{4 \cdot \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      12. *-lft-identity N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{1 \cdot z}}{y}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      13. associate-*l/ N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{y} \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(\frac{1}{y} \cdot z\right)\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      15. associate-*l* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot \frac{1}{y}\right) \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      16. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot \left(4 \cdot \frac{1}{y}\right)}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      17. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(z \cdot \left(4 \cdot \frac{1}{y}\right)\right)\right) + \color{blue}{3}\right) + 1 \]

      18. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z \cdot \left(4 \cdot \frac{1}{y}\right)\right)\right) + \left(3 + 1\right)} \]

      19. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(z \cdot \left(4 \cdot \frac{1}{y}\right)\right)\right) + \color{blue}{4} \]

   5. Applied rewrites 87.1%

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

Alternative 9: 99.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x - z, \frac{4}{y}, 4\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (- x z) (/ 4.0 y) 4.0))

C

double code(double x, double y, double z) {
	return fma((x - z), (4.0 / y), 4.0);
}

Julia

function code(x, y, z)
	return fma(Float64(x - z), Float64(4.0 / y), 4.0)
end

Wolfram

code[x_, y_, z_] := N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision] + 4.0), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + \left(4 \cdot \frac{x}{y} + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}\right)} \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x - z, \frac{4}{y}, 4\right)} \]
6. Add Preprocessing

Alternative 10: 34.1% accurate, 31.0× speedup

Math

\[\begin{array}{l} \\ 4 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 4.0)

C

double code(double x, double y, double z) {
	return 4.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 4.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 4.0;
}

Python

def code(x, y, z):
	return 4.0

Julia

function code(x, y, z)
	return 4.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 4.0;
end

Wolfram

code[x_, y_, z_] := 4.0

Derivation

1. Initial program 99.5%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

5. Applied rewrites 37.0%

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

Reproduce

herbie shell --seed 2024294 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))