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

Percentage Accurate: 99.8% → 100.0%

Time: 6.0s

Alternatives: 7

Speedup: 1.5×

• 21.4% 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: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. +-commutative N/A

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

   4. div-add-rev N/A

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

   5. associate-*r/ N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. associate-+r+ N/A

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

   9. sub-neg N/A

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

   10. distribute-lft-in N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. +-commutative N/A

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

   14. div-add N/A

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

   15. associate-*r/ N/A

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

5. Applied rewrites 100.0%

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

Alternative 2: 66.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} \cdot 4\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+283}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ x y) 4.0)) (t_1 (/ (* (- (+ (* 0.75 y) x) z) 4.0) y)))
   (if (<= t_1 -5.0)
     t_0
     (if (<= t_1 5.0) 4.0 (if (<= t_1 2e+283) t_0 (* (/ z y) -4.0))))))

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 tmp;
	if (t_1 <= -5.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else if (t_1 <= 2e+283) {
		tmp = t_0;
	} else {
		tmp = (z / y) * -4.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x / y) * 4.0d0
    t_1 = ((((0.75d0 * y) + x) - z) * 4.0d0) / y
    if (t_1 <= (-5.0d0)) then
        tmp = t_0
    else if (t_1 <= 5.0d0) then
        tmp = 4.0d0
    else if (t_1 <= 2d+283) then
        tmp = t_0
    else
        tmp = (z / y) * (-4.0d0)
    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 tmp;
	if (t_1 <= -5.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else if (t_1 <= 2e+283) {
		tmp = t_0;
	} else {
		tmp = (z / y) * -4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x / y) * 4.0
	t_1 = ((((0.75 * y) + x) - z) * 4.0) / y
	tmp = 0
	if t_1 <= -5.0:
		tmp = t_0
	elif t_1 <= 5.0:
		tmp = 4.0
	elif t_1 <= 2e+283:
		tmp = t_0
	else:
		tmp = (z / y) * -4.0
	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)
	tmp = 0.0
	if (t_1 <= -5.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	elseif (t_1 <= 2e+283)
		tmp = t_0;
	else
		tmp = Float64(Float64(z / y) * -4.0);
	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;
	tmp = 0.0;
	if (t_1 <= -5.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	elseif (t_1 <= 2e+283)
		tmp = t_0;
	else
		tmp = (z / y) * -4.0;
	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]}, If[LessEqual[t$95$1, -5.0], t$95$0, If[LessEqual[t$95$1, 5.0], 4.0, If[LessEqual[t$95$1, 2e+283], t$95$0, N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]]]]]]

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) < -5 or 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 1.99999999999999991e283

   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

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. +-commutative N/A

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

      4. div-add-rev N/A

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

      5. associate-*r/ N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. div-add N/A

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

      15. associate-*r/ N/A

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

   5. Applied rewrites 100.0%

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

   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 62.9

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

   8. Applied rewrites 62.9%

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

   if -5 < (/.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 96.9%

      \[\leadsto \color{blue}{4} \]

   if 1.99999999999999991e283 < (/.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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. associate--r+ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. *-commutative N/A

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

      16. neg-mul-1 N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. associate-*l/ N/A

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

      21. associate-*l* N/A

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

      22. lower-*.f64 N/A

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 65.9%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -5:\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 5:\\ \;\;\;\;4\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - z}{y} \cdot 4\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 50000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -5.0) t_0 (if (<= t_1 50000000.0) (fma (/ 4.0 y) x 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 <= -5.0) {
		tmp = t_0;
	} else if (t_1 <= 50000000.0) {
		tmp = fma((4.0 / y), x, 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 <= -5.0)
		tmp = t_0;
	elseif (t_1 <= 50000000.0)
		tmp = fma(Float64(4.0 / y), x, 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, -5.0], t$95$0, If[LessEqual[t$95$1, 50000000.0], N[(N[(4.0 / y), $MachinePrecision] * x + 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) < -5 or 5e7 < (/.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

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. +-commutative N/A

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

      4. div-add-rev N/A

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

      5. associate-*r/ N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. div-add N/A

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

      15. associate-*r/ N/A

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

   5. Applied rewrites 100.0%

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

   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 98.7

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

   8. Applied rewrites 98.7%

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

   if -5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5e7

   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

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

      1. *-commutative N/A

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

      2. div-add N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. associate-+r+ N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. *-lft-identity N/A

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

      14. associate-*l/ N/A

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

      15. associate-*l* N/A

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

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(4 \cdot \frac{1}{y}, x, 4\right)} \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4 \cdot 1}{y}}, x, 4\right) \]

      18. metadata-eval N/A

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

      19. lower-/.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -5:\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 50000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z}{y} \cdot -4\\ t_1 := \frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y}\\ \mathbf{if}\;t\_1 \leq -5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 50000000:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ z y) -4.0)) (t_1 (/ (* (- (+ (* 0.75 y) x) z) 4.0) y)))
   (if (<= t_1 -5.0) t_0 (if (<= t_1 50000000.0) 4.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z / y) * -4.0;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -5.0) {
		tmp = t_0;
	} else if (t_1 <= 50000000.0) {
		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 = (z / y) * (-4.0d0)
    t_1 = ((((0.75d0 * y) + x) - z) * 4.0d0) / y
    if (t_1 <= (-5.0d0)) then
        tmp = t_0
    else if (t_1 <= 50000000.0d0) 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 = (z / y) * -4.0;
	double t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	double tmp;
	if (t_1 <= -5.0) {
		tmp = t_0;
	} else if (t_1 <= 50000000.0) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z / y) * -4.0
	t_1 = ((((0.75 * y) + x) - z) * 4.0) / y
	tmp = 0
	if t_1 <= -5.0:
		tmp = t_0
	elif t_1 <= 50000000.0:
		tmp = 4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(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 <= -5.0)
		tmp = t_0;
	elseif (t_1 <= 50000000.0)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z / y) * -4.0;
	t_1 = ((((0.75 * y) + x) - z) * 4.0) / y;
	tmp = 0.0;
	if (t_1 <= -5.0)
		tmp = t_0;
	elseif (t_1 <= 50000000.0)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z / 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, -5.0], t$95$0, If[LessEqual[t$95$1, 50000000.0], 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) < -5 or 5e7 < (/.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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. associate--r+ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. *-commutative N/A

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

      16. neg-mul-1 N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. associate-*l/ N/A

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

      21. associate-*l* N/A

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

      22. lower-*.f64 N/A

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

   5. Applied rewrites 47.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 45.9%

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

   if -5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5e7

   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 95.1%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq -5:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{elif}\;\frac{\left(\left(0.75 \cdot y + x\right) - z\right) \cdot 4}{y} \leq 50000000:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-81}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ 4.0 y) x 4.0)))
   (if (<= x -1e-9) t_0 (if (<= x 3e-81) (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 <= -1e-9) {
		tmp = t_0;
	} else if (x <= 3e-81) {
		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 <= -1e-9)
		tmp = t_0;
	elseif (x <= 3e-81)
		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, -1e-9], t$95$0, If[LessEqual[x, 3e-81], N[(-4.0 * N[(z / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000006e-9 or 2.9999999999999999e-81 < 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

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

      1. *-commutative N/A

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

      2. div-add N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. associate-+r+ N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. *-lft-identity N/A

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

      14. associate-*l/ N/A

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

      15. associate-*l* N/A

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

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(4 \cdot \frac{1}{y}, x, 4\right)} \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4 \cdot 1}{y}}, x, 4\right) \]

      18. metadata-eval N/A

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

      19. lower-/.f64 87.8

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

   5. Applied rewrites 87.8%

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

   if -1.00000000000000006e-9 < x < 2.9999999999999999e-81

   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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. associate--r+ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. *-commutative N/A

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

      16. neg-mul-1 N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. associate-*l/ N/A

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

      21. associate-*l* N/A

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

      22. lower-*.f64 N/A

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

   5. Applied rewrites 96.6%

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

   7. Applied rewrites 96.8%

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

Alternative 6: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} \cdot 4\\ \mathbf{if}\;x \leq -4 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ x y) 4.0)))
   (if (<= x -4e+23) t_0 (if (<= x 4.8e+115) (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 <= -4e+23) {
		tmp = t_0;
	} else if (x <= 4.8e+115) {
		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 <= -4e+23)
		tmp = t_0;
	elseif (x <= 4.8e+115)
		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, -4e+23], t$95$0, If[LessEqual[x, 4.8e+115], N[(-4.0 * N[(z / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.9999999999999997e23 or 4.8000000000000001e115 < 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

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. +-commutative N/A

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

      4. div-add-rev N/A

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

      5. associate-*r/ N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. div-add N/A

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

      15. associate-*r/ N/A

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

   5. Applied rewrites 100.0%

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

   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 75.8

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

   8. Applied rewrites 75.8%

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

   if -3.9999999999999997e23 < x < 4.8000000000000001e115

   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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. associate--r+ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. *-commutative N/A

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

      16. neg-mul-1 N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. associate-*l/ N/A

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

      21. associate-*l* N/A

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

      22. lower-*.f64 N/A

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 84.5%

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

Alternative 7: 34.0% 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 36.0%

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

Reproduce

herbie shell --seed 2024298 
(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)))