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

Percentage Accurate: 99.8% → 99.8%

Time: 7.9s

Alternatives: 9

Speedup: 1.0×

• 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: 99.8% 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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 67.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} \cdot 4\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ t_2 := \frac{-4 \cdot z}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+139}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -10:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+23}:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+233}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ x y) 4.0))
        (t_1 (/ (* (- (+ (* 0.75 y) x) z) 4.0) y))
        (t_2 (/ (* -4.0 z) y)))
   (if (<= t_1 -2e+139)
     t_0
     (if (<= t_1 -10.0)
       t_2
       (if (<= t_1 2e+23)
         4.0
         (if (<= t_1 1e+77) t_0 (if (<= t_1 1e+233) t_2 t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x / y) * 4.0;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double t_2 = (-4.0 * z) / y;
	double tmp;
	if (t_1 <= -2e+139) {
		tmp = t_0;
	} else if (t_1 <= -10.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+23) {
		tmp = 4.0;
	} else if (t_1 <= 1e+77) {
		tmp = t_0;
	} else if (t_1 <= 1e+233) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = (x / y) * 4.0d0
    t_1 = ((((0.75d0 * y) + x) - z) * 4.0d0) / y
    t_2 = ((-4.0d0) * z) / y
    if (t_1 <= (-2d+139)) then
        tmp = t_0
    else if (t_1 <= (-10.0d0)) then
        tmp = t_2
    else if (t_1 <= 2d+23) then
        tmp = 4.0d0
    else if (t_1 <= 1d+77) then
        tmp = t_0
    else if (t_1 <= 1d+233) then
        tmp = t_2
    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 / y) * 4.0;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double t_2 = (-4.0 * z) / y;
	double tmp;
	if (t_1 <= -2e+139) {
		tmp = t_0;
	} else if (t_1 <= -10.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+23) {
		tmp = 4.0;
	} else if (t_1 <= 1e+77) {
		tmp = t_0;
	} else if (t_1 <= 1e+233) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x / y) * 4.0
	t_1 = ((((0.75 * y) + x) - z) * 4.0) / y
	t_2 = (-4.0 * z) / y
	tmp = 0
	if t_1 <= -2e+139:
		tmp = t_0
	elif t_1 <= -10.0:
		tmp = t_2
	elif t_1 <= 2e+23:
		tmp = 4.0
	elif t_1 <= 1e+77:
		tmp = t_0
	elif t_1 <= 1e+233:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / y) * 4.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(0.75 * y) + x) - z) * 4.0) / y)
	t_2 = Float64(Float64(-4.0 * z) / y)
	tmp = 0.0
	if (t_1 <= -2e+139)
		tmp = t_0;
	elseif (t_1 <= -10.0)
		tmp = t_2;
	elseif (t_1 <= 2e+23)
		tmp = 4.0;
	elseif (t_1 <= 1e+77)
		tmp = t_0;
	elseif (t_1 <= 1e+233)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x / y) * 4.0;
	t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	t_2 = (-4.0 * z) / y;
	tmp = 0.0;
	if (t_1 <= -2e+139)
		tmp = t_0;
	elseif (t_1 <= -10.0)
		tmp = t_2;
	elseif (t_1 <= 2e+23)
		tmp = 4.0;
	elseif (t_1 <= 1e+77)
		tmp = t_0;
	elseif (t_1 <= 1e+233)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / 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]}, Block[{t$95$2 = N[(N[(-4.0 * z), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+139], t$95$0, If[LessEqual[t$95$1, -10.0], t$95$2, If[LessEqual[t$95$1, 2e+23], 4.0, If[LessEqual[t$95$1, 1e+77], t$95$0, If[LessEqual[t$95$1, 1e+233], t$95$2, t$95$0]]]]]]]]

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) < -2.00000000000000007e139 or 1.9999999999999998e23 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 9.99999999999999983e76 or 9.99999999999999974e232 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   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 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 99.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 64.8

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

   8. Applied rewrites 64.8%

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

   if -2.00000000000000007e139 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -10 or 9.99999999999999983e76 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 9.99999999999999974e232

   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 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 64.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 62.5%

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

   10. Applied rewrites 62.8%

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

   if -10 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 1.9999999999999998e23

   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 97.0%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -2 \cdot 10^{+139}:\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -10:\\ \;\;\;\;\frac{-4 \cdot z}{y}\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 2 \cdot 10^{+23}:\\ \;\;\;\;4\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 10^{+77}:\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 10^{+233}:\\ \;\;\;\;\frac{-4 \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - z}{y} \cdot 4\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (- x z) y) 4.0)) (t_1 (/ (* (- (+ (* 0.75 y) x) z) 4.0) y)))
   (if (<= t_1 -1e+26) t_0 (if (<= t_1 5.0) (fma -4.0 (/ z 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 <= -1e+26) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = fma(-4.0, (z / 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 <= -1e+26)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		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[(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, -1e+26], t$95$0, If[LessEqual[t$95$1, 5.0], N[(-4.0 * N[(z / 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) < -1.00000000000000005e26 or 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   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 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 99.6%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -1.00000000000000005e26 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5

   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 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 99.1%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -1 \cdot 10^{+26}:\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 5:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4 \cdot z}{y}\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -10:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -10.0) t_0 (if (<= t_1 5.0) 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 <= -10.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		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 <= (-10.0d0)) then
        tmp = t_0
    else if (t_1 <= 5.0d0) 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 <= -10.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		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 <= -10.0:
		tmp = t_0
	elif t_1 <= 5.0:
		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 <= -10.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		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 <= -10.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		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, -10.0], t$95$0, If[LessEqual[t$95$1, 5.0], 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) < -10 or 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   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 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 52.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 51.0%

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

   10. Applied rewrites 51.2%

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

   if -10 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5

   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 98.0%

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

4. Final simplification 67.1%

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

Alternative 5: 67.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4}{y} \cdot z\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -10:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -10.0) t_0 (if (<= t_1 5.0) 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 <= -10.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		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 <= (-10.0d0)) then
        tmp = t_0
    else if (t_1 <= 5.0d0) 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 <= -10.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		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 <= -10.0:
		tmp = t_0
	elif t_1 <= 5.0:
		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 <= -10.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		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 <= -10.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		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, -10.0], t$95$0, If[LessEqual[t$95$1, 5.0], 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) < -10 or 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   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 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 52.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 51.0%

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

   if -10 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5

   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 98.0%

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

4. Final simplification 67.0%

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

Alternative 6: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+122}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ 4.0 y) x 4.0)))
   (if (<= x -4.5e+122) t_0 (if (<= x 4e+17) (fma -4.0 (/ z y) 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((4.0 / y), x, 4.0);
	double tmp;
	if (x <= -4.5e+122) {
		tmp = t_0;
	} else if (x <= 4e+17) {
		tmp = fma(-4.0, (z / y), 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(4.0 / y), x, 4.0)
	tmp = 0.0
	if (x <= -4.5e+122)
		tmp = t_0;
	elseif (x <= 4e+17)
		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[(4.0 / y), $MachinePrecision] * x + 4.0), $MachinePrecision]}, If[LessEqual[x, -4.5e+122], t$95$0, If[LessEqual[x, 4e+17], N[(-4.0 * N[(z / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.49999999999999997e122 or 4e17 < x

   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 z around 0

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

   5. Applied rewrites 93.6%

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

   7. Applied rewrites 93.6%

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

   if -4.49999999999999997e122 < x < 4e17

   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 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 90.1%

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

Alternative 7: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} \cdot 4\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{+122}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ x y) 4.0)))
   (if (<= x -5.4e+122) t_0 (if (<= x 3.8e+64) (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 <= -5.4e+122) {
		tmp = t_0;
	} else if (x <= 3.8e+64) {
		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 <= -5.4e+122)
		tmp = t_0;
	elseif (x <= 3.8e+64)
		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, -5.4e+122], t$95$0, If[LessEqual[x, 3.8e+64], N[(-4.0 * N[(z / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.3999999999999997e122 or 3.8000000000000001e64 < x

   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 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. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 81.6

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

   8. Applied rewrites 81.6%

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

   if -5.3999999999999997e122 < x < 3.8000000000000001e64

   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 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 88.8%

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

Alternative 8: 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 99.9%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
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 9: 35.4% 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 99.9%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 35.2%

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

Reproduce

herbie shell --seed 2024331 
(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)))