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

Percentage Accurate: 99.8% → 100.0%

Time: 7.5s

Alternatives: 9

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

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

   2. associate-+l+ N/A

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

   3. div-sub N/A

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

   4. associate-/l* N/A

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

   5. *-inverses N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. sub-neg N/A

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

   10. distribute-rgt-in N/A

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

   11. metadata-eval N/A

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

   12. associate-+r+ N/A

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

   13. metadata-eval N/A

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

   14. distribute-rgt1-in N/A

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

   15. distribute-rgt-in N/A

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

   16. associate-+l+ N/A

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

5. Applied rewrites 100.0%

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

Alternative 2: 67.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} \cdot 4\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -4000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ x y) 4.0)) (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (<= t_1 -4000.0)
     t_0
     (if (<= t_1 5.0) 4.0 (if (<= t_1 2e+272) 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 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -4000.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else if (t_1 <= 2e+272) {
		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 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    if (t_1 <= (-4000.0d0)) then
        tmp = t_0
    else if (t_1 <= 5.0d0) then
        tmp = 4.0d0
    else if (t_1 <= 2d+272) 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 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -4000.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else if (t_1 <= 2e+272) {
		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 = (4.0 * ((x + (y * 0.75)) - z)) / y
	tmp = 0
	if t_1 <= -4000.0:
		tmp = t_0
	elif t_1 <= 5.0:
		tmp = 4.0
	elif t_1 <= 2e+272:
		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(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -4000.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	elseif (t_1 <= 2e+272)
		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 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	tmp = 0.0;
	if (t_1 <= -4000.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	elseif (t_1 <= 2e+272)
		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[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -4000.0], t$95$0, If[LessEqual[t$95$1, 5.0], 4.0, If[LessEqual[t$95$1, 2e+272], 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) < -4e3 or 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 2.0000000000000001e272

   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 inf

      \[\leadsto \color{blue}{4 \cdot \frac{x}{y}} \]
   4. 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 55.6

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

   5. Applied rewrites 55.6%

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

   if -4e3 < (/.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 99.0%

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

   if 2.0000000000000001e272 < (/.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 y around 0

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

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.1%

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

Alternative 3: 67.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{4}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -4000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ 4.0 y))) (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (<= t_1 -4000.0)
     t_0
     (if (<= t_1 5.0) 4.0 (if (<= t_1 2e+272) t_0 (* (/ z y) -4.0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -4000.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else if (t_1 <= 2e+272) {
		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 * (4.0d0 / y)
    t_1 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    if (t_1 <= (-4000.0d0)) then
        tmp = t_0
    else if (t_1 <= 5.0d0) then
        tmp = 4.0d0
    else if (t_1 <= 2d+272) 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 * (4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -4000.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else if (t_1 <= 2e+272) {
		tmp = t_0;
	} else {
		tmp = (z / y) * -4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (4.0 / y)
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y
	tmp = 0
	if t_1 <= -4000.0:
		tmp = t_0
	elif t_1 <= 5.0:
		tmp = 4.0
	elif t_1 <= 2e+272:
		tmp = t_0
	else:
		tmp = (z / y) * -4.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(4.0 / y))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -4000.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	elseif (t_1 <= 2e+272)
		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 * (4.0 / y);
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	tmp = 0.0;
	if (t_1 <= -4000.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	elseif (t_1 <= 2e+272)
		tmp = t_0;
	else
		tmp = (z / y) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -4000.0], t$95$0, If[LessEqual[t$95$1, 5.0], 4.0, If[LessEqual[t$95$1, 2e+272], 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) < -4e3 or 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 2.0000000000000001e272

   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 inf

      \[\leadsto \color{blue}{4 \cdot \frac{x}{y}} \]
   4. 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 55.6

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 55.5%

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

   if -4e3 < (/.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 99.0%

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

   if 2.0000000000000001e272 < (/.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 y around 0

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

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.1%

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

Alternative 4: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_0 \leq -1000000000 \lor \neg \left(t\_0 \leq 1000000000000\right):\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (or (<= t_0 -1000000000.0) (not (<= t_0 1000000000000.0)))
     (* (/ (- x z) y) 4.0)
     (fma (/ x y) 4.0 4.0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if ((t_0 <= -1000000000.0) || !(t_0 <= 1000000000000.0)) {
		tmp = ((x - z) / y) * 4.0;
	} else {
		tmp = fma((x / y), 4.0, 4.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if ((t_0 <= -1000000000.0) || !(t_0 <= 1000000000000.0))
		tmp = Float64(Float64(Float64(x - z) / y) * 4.0);
	else
		tmp = fma(Float64(x / y), 4.0, 4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1000000000.0], N[Not[LessEqual[t$95$0, 1000000000000.0]], $MachinePrecision]], N[(N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] * 4.0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * 4.0 + 4.0), $MachinePrecision]]]

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) < -1e9 or 1e12 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - z}{y}} \]
   4. 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 98.8

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

   5. Applied rewrites 98.8%

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

   if -1e9 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 1e12

   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

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. sub-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

      12. associate-+r+ N/A

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

      13. metadata-eval N/A

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

      14. distribute-rgt1-in N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-+l+ N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 99.5%

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

   10. Applied rewrites 99.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -1000000000 \lor \neg \left(\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 1000000000000\right):\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_0 \leq -4000 \lor \neg \left(t\_0 \leq 1000000000000\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (or (<= t_0 -4000.0) (not (<= t_0 1000000000000.0)))
     (* (/ z y) -4.0)
     4.0)))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if ((t_0 <= -4000.0) || !(t_0 <= 1000000000000.0)) {
		tmp = (z / y) * -4.0;
	} else {
		tmp = 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) :: tmp
    t_0 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    if ((t_0 <= (-4000.0d0)) .or. (.not. (t_0 <= 1000000000000.0d0))) then
        tmp = (z / y) * (-4.0d0)
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if ((t_0 <= -4000.0) || !(t_0 <= 1000000000000.0)) {
		tmp = (z / y) * -4.0;
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y
	tmp = 0
	if (t_0 <= -4000.0) or not (t_0 <= 1000000000000.0):
		tmp = (z / y) * -4.0
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if ((t_0 <= -4000.0) || !(t_0 <= 1000000000000.0))
		tmp = Float64(Float64(z / y) * -4.0);
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	tmp = 0.0;
	if ((t_0 <= -4000.0) || ~((t_0 <= 1000000000000.0)))
		tmp = (z / y) * -4.0;
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4000.0], N[Not[LessEqual[t$95$0, 1000000000000.0]], $MachinePrecision]], N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision], 4.0]]

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) < -4e3 or 1e12 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - z}{y}} \]
   4. 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 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.4%

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

   if -4e3 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 1e12

   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 2 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -4000 \lor \neg \left(\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 1000000000000\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+93} \lor \neg \left(z \leq 29000000000\right):\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.25e+93) (not (<= z 29000000000.0)))
   (fma -4.0 (/ z y) 4.0)
   (fma (/ x y) 4.0 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.25e+93) || !(z <= 29000000000.0)) {
		tmp = fma(-4.0, (z / y), 4.0);
	} else {
		tmp = fma((x / y), 4.0, 4.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.25e+93) || !(z <= 29000000000.0))
		tmp = fma(-4.0, Float64(z / y), 4.0);
	else
		tmp = fma(Float64(x / y), 4.0, 4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.25e+93], N[Not[LessEqual[z, 29000000000.0]], $MachinePrecision]], N[(-4.0 * N[(z / y), $MachinePrecision] + 4.0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * 4.0 + 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25e93 or 2.9e10 < z

   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. div-sub N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. *-lft-identity N/A

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

      11. associate-*l/ N/A

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

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

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

      13. associate-*l* N/A

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

      14. metadata-eval N/A

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

      15. associate-+l+ N/A

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

      16. metadata-eval N/A

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

   5. Applied rewrites 90.4%

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

   if -1.25e93 < z < 2.9e10

   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

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

      2. associate-+l+ N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. sub-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

      12. associate-+r+ N/A

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

      13. metadata-eval N/A

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

      14. distribute-rgt1-in N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-+l+ N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 92.3%

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

   10. Applied rewrites 92.4%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+93} \lor \neg \left(z \leq 29000000000\right):\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.05 \cdot 10^{+93} \lor \neg \left(z \leq 5 \cdot 10^{+88}\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.05e+93) (not (<= z 5e+88)))
   (* (/ z y) -4.0)
   (fma (/ x y) 4.0 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.05e+93) || !(z <= 5e+88)) {
		tmp = (z / y) * -4.0;
	} else {
		tmp = fma((x / y), 4.0, 4.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.05e+93) || !(z <= 5e+88))
		tmp = Float64(Float64(z / y) * -4.0);
	else
		tmp = fma(Float64(x / y), 4.0, 4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.05e+93], N[Not[LessEqual[z, 5e+88]], $MachinePrecision]], N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * 4.0 + 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.05e93 or 4.99999999999999997e88 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - z}{y}} \]
   4. 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 86.5

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 78.3%

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

   if -3.05e93 < z < 4.99999999999999997e88

   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

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

      2. associate-+l+ N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. sub-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

      12. associate-+r+ N/A

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

      13. metadata-eval N/A

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

      14. distribute-rgt1-in N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-+l+ N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 89.2%

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

   10. Applied rewrites 89.3%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.05 \cdot 10^{+93} \lor \neg \left(z \leq 5 \cdot 10^{+88}\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, 4, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.05 \cdot 10^{+93} \lor \neg \left(z \leq 5 \cdot 10^{+88}\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.05e+93) (not (<= z 5e+88)))
   (* (/ z y) -4.0)
   (fma (/ 4.0 y) x 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.05e+93) || !(z <= 5e+88)) {
		tmp = (z / y) * -4.0;
	} else {
		tmp = fma((4.0 / y), x, 4.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.05e+93) || !(z <= 5e+88))
		tmp = Float64(Float64(z / y) * -4.0);
	else
		tmp = fma(Float64(4.0 / y), x, 4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.05e+93], N[Not[LessEqual[z, 5e+88]], $MachinePrecision]], N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(4.0 / y), $MachinePrecision] * x + 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.05e93 or 4.99999999999999997e88 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - z}{y}} \]
   4. 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 86.5

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 78.3%

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

   if -3.05e93 < z < 4.99999999999999997e88

   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

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

      2. associate-+l+ N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. sub-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

      12. associate-+r+ N/A

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

      13. metadata-eval N/A

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

      14. distribute-rgt1-in N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-+l+ N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 89.2%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.05 \cdot 10^{+93} \lor \neg \left(z \leq 5 \cdot 10^{+88}\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 34.2%

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

Reproduce

herbie shell --seed 2024318 
(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)))