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

Percentage Accurate: 99.7% → 99.7%

Time: 6.4s

Alternatives: 9

Speedup: 1.1×

• 20.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return fma((fma(0.75, y, x) - z), (4.0 / y), 1.0);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 99.7

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.75, y, x\right) - z, \color{blue}{\frac{4}{y}}, 1\right) \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.75, y, x\right) - z, \frac{4}{y}, 1\right)} \]
5. Add Preprocessing

Alternative 2: 65.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z}{y} \cdot -4\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ t_2 := \frac{x}{y} \cdot 4\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+301}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+61}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+27}:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 10^{+219}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ z y) -4.0))
        (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y))
        (t_2 (* (/ x y) 4.0)))
   (if (<= t_1 -5e+301)
     t_0
     (if (<= t_1 -1e+61)
       t_2
       (if (<= t_1 -2e+16)
         t_0
         (if (<= t_1 2e+27) 4.0 (if (<= t_1 1e+219) t_2 t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = (z / y) * -4.0;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_2 = (x / y) * 4.0;
	double tmp;
	if (t_1 <= -5e+301) {
		tmp = t_0;
	} else if (t_1 <= -1e+61) {
		tmp = t_2;
	} else if (t_1 <= -2e+16) {
		tmp = t_0;
	} else if (t_1 <= 2e+27) {
		tmp = 4.0;
	} else if (t_1 <= 1e+219) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (z / y) * (-4.0d0)
    t_1 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    t_2 = (x / y) * 4.0d0
    if (t_1 <= (-5d+301)) then
        tmp = t_0
    else if (t_1 <= (-1d+61)) then
        tmp = t_2
    else if (t_1 <= (-2d+16)) then
        tmp = t_0
    else if (t_1 <= 2d+27) then
        tmp = 4.0d0
    else if (t_1 <= 1d+219) then
        tmp = t_2
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z / y) * -4.0;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_2 = (x / y) * 4.0;
	double tmp;
	if (t_1 <= -5e+301) {
		tmp = t_0;
	} else if (t_1 <= -1e+61) {
		tmp = t_2;
	} else if (t_1 <= -2e+16) {
		tmp = t_0;
	} else if (t_1 <= 2e+27) {
		tmp = 4.0;
	} else if (t_1 <= 1e+219) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z / y) * -4.0
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y
	t_2 = (x / y) * 4.0
	tmp = 0
	if t_1 <= -5e+301:
		tmp = t_0
	elif t_1 <= -1e+61:
		tmp = t_2
	elif t_1 <= -2e+16:
		tmp = t_0
	elif t_1 <= 2e+27:
		tmp = 4.0
	elif t_1 <= 1e+219:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z / y) * -4.0)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	t_2 = Float64(Float64(x / y) * 4.0)
	tmp = 0.0
	if (t_1 <= -5e+301)
		tmp = t_0;
	elseif (t_1 <= -1e+61)
		tmp = t_2;
	elseif (t_1 <= -2e+16)
		tmp = t_0;
	elseif (t_1 <= 2e+27)
		tmp = 4.0;
	elseif (t_1 <= 1e+219)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z / y) * -4.0;
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	t_2 = (x / y) * 4.0;
	tmp = 0.0;
	if (t_1 <= -5e+301)
		tmp = t_0;
	elseif (t_1 <= -1e+61)
		tmp = t_2;
	elseif (t_1 <= -2e+16)
		tmp = t_0;
	elseif (t_1 <= 2e+27)
		tmp = 4.0;
	elseif (t_1 <= 1e+219)
		tmp = t_2;
	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[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+301], t$95$0, If[LessEqual[t$95$1, -1e+61], t$95$2, If[LessEqual[t$95$1, -2e+16], t$95$0, If[LessEqual[t$95$1, 2e+27], 4.0, If[LessEqual[t$95$1, 1e+219], t$95$2, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -5.0000000000000004e301 or -9.99999999999999949e60 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -2e16 or 9.99999999999999965e218 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 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%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.3%

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

   if -5.0000000000000004e301 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -9.99999999999999949e60 or 2e27 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 9.99999999999999965e218

   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 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 63.8

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

   5. Applied rewrites 63.8%

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

   if -2e16 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 2e27

   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 inf

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

   5. Applied rewrites 95.0%

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

Alternative 3: 65.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z}{y} \cdot -4\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ t_2 := x \cdot \frac{4}{y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+301}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+61}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+27}:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 10^{+219}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ z y) -4.0))
        (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y))
        (t_2 (* x (/ 4.0 y))))
   (if (<= t_1 -5e+301)
     t_0
     (if (<= t_1 -1e+61)
       t_2
       (if (<= t_1 -2e+16)
         t_0
         (if (<= t_1 2e+27) 4.0 (if (<= t_1 1e+219) t_2 t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = (z / y) * -4.0;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_2 = x * (4.0 / y);
	double tmp;
	if (t_1 <= -5e+301) {
		tmp = t_0;
	} else if (t_1 <= -1e+61) {
		tmp = t_2;
	} else if (t_1 <= -2e+16) {
		tmp = t_0;
	} else if (t_1 <= 2e+27) {
		tmp = 4.0;
	} else if (t_1 <= 1e+219) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (z / y) * (-4.0d0)
    t_1 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    t_2 = x * (4.0d0 / y)
    if (t_1 <= (-5d+301)) then
        tmp = t_0
    else if (t_1 <= (-1d+61)) then
        tmp = t_2
    else if (t_1 <= (-2d+16)) then
        tmp = t_0
    else if (t_1 <= 2d+27) then
        tmp = 4.0d0
    else if (t_1 <= 1d+219) then
        tmp = t_2
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z / y) * -4.0;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double t_2 = x * (4.0 / y);
	double tmp;
	if (t_1 <= -5e+301) {
		tmp = t_0;
	} else if (t_1 <= -1e+61) {
		tmp = t_2;
	} else if (t_1 <= -2e+16) {
		tmp = t_0;
	} else if (t_1 <= 2e+27) {
		tmp = 4.0;
	} else if (t_1 <= 1e+219) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z / y) * -4.0
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y
	t_2 = x * (4.0 / y)
	tmp = 0
	if t_1 <= -5e+301:
		tmp = t_0
	elif t_1 <= -1e+61:
		tmp = t_2
	elif t_1 <= -2e+16:
		tmp = t_0
	elif t_1 <= 2e+27:
		tmp = 4.0
	elif t_1 <= 1e+219:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z / y) * -4.0)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	t_2 = Float64(x * Float64(4.0 / y))
	tmp = 0.0
	if (t_1 <= -5e+301)
		tmp = t_0;
	elseif (t_1 <= -1e+61)
		tmp = t_2;
	elseif (t_1 <= -2e+16)
		tmp = t_0;
	elseif (t_1 <= 2e+27)
		tmp = 4.0;
	elseif (t_1 <= 1e+219)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z / y) * -4.0;
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	t_2 = x * (4.0 / y);
	tmp = 0.0;
	if (t_1 <= -5e+301)
		tmp = t_0;
	elseif (t_1 <= -1e+61)
		tmp = t_2;
	elseif (t_1 <= -2e+16)
		tmp = t_0;
	elseif (t_1 <= 2e+27)
		tmp = 4.0;
	elseif (t_1 <= 1e+219)
		tmp = t_2;
	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[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+301], t$95$0, If[LessEqual[t$95$1, -1e+61], t$95$2, If[LessEqual[t$95$1, -2e+16], t$95$0, If[LessEqual[t$95$1, 2e+27], 4.0, If[LessEqual[t$95$1, 1e+219], t$95$2, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -5.0000000000000004e301 or -9.99999999999999949e60 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -2e16 or 9.99999999999999965e218 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 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%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.3%

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

   if -5.0000000000000004e301 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -9.99999999999999949e60 or 2e27 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 9.99999999999999965e218

   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 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 63.8

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

   5. Applied rewrites 63.8%

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

   7. Applied rewrites 63.6%

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

   if -2e16 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 2e27

   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 inf

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

   5. Applied rewrites 95.0%

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

Alternative 4: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+16} \lor \neg \left(t\_0 \leq 20000000000\right):\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(\frac{-4}{y}, z, 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (or (<= t_0 -2e+16) (not (<= t_0 20000000000.0)))
     (* (/ (- x z) y) 4.0)
     (+ 1.0 (fma (/ -4.0 y) z 3.0)))))

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 <= -2e+16) || !(t_0 <= 20000000000.0)) {
		tmp = ((x - z) / y) * 4.0;
	} else {
		tmp = 1.0 + fma((-4.0 / y), z, 3.0);
	}
	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 <= -2e+16) || !(t_0 <= 20000000000.0))
		tmp = Float64(Float64(Float64(x - z) / y) * 4.0);
	else
		tmp = Float64(1.0 + fma(Float64(-4.0 / y), z, 3.0));
	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, -2e+16], N[Not[LessEqual[t$95$0, 20000000000.0]], $MachinePrecision]], N[(N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] * 4.0), $MachinePrecision], N[(1.0 + N[(N[(-4.0 / y), $MachinePrecision] * z + 3.0), $MachinePrecision]), $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) < -2e16 or 2e10 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - z}{y}} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   if -2e16 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 2e10

   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

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

      1. associate-*r/ N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. div-add N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. associate-/l* N/A

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

      17. *-inverses N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 N/A

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

      20. metadata-eval N/A

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

      21. associate-*r/ N/A

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

      22. metadata-eval N/A

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

      23. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

      \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\frac{-4}{y}, z, 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -2 \cdot 10^{+16} \lor \neg \left(\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 20000000000\right):\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;1 + \mathsf{fma}\left(\frac{-4}{y}, z, 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+21} \lor \neg \left(t\_0 \leq 5\right):\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (or (<= t_0 -5e+21) (not (<= t_0 5.0)))
     (* (/ (- x z) y) 4.0)
     (fma (/ 4.0 y) x 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 <= -5e+21) || !(t_0 <= 5.0)) {
		tmp = ((x - z) / y) * 4.0;
	} else {
		tmp = fma((4.0 / y), x, 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 <= -5e+21) || !(t_0 <= 5.0))
		tmp = Float64(Float64(Float64(x - z) / y) * 4.0);
	else
		tmp = fma(Float64(4.0 / y), x, 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, -5e+21], N[Not[LessEqual[t$95$0, 5.0]], $MachinePrecision]], N[(N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] * 4.0), $MachinePrecision], N[(N[(4.0 / y), $MachinePrecision] * x + 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) < -5e21 or 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - z}{y}} \]
   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 99.2

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

   5. Applied rewrites 99.2%

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

   if -5e21 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5

   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. div-add N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-lft-identity N/A

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

      13. associate-*l/ N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -5 \cdot 10^{+21} \lor \neg \left(\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 5\right):\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+21} \lor \neg \left(t\_0 \leq 5\right):\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (or (<= t_0 -5e+21) (not (<= t_0 5.0)))
     (* (- x z) (/ 4.0 y))
     (fma (/ 4.0 y) x 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 <= -5e+21) || !(t_0 <= 5.0)) {
		tmp = (x - z) * (4.0 / y);
	} else {
		tmp = fma((4.0 / y), x, 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 <= -5e+21) || !(t_0 <= 5.0))
		tmp = Float64(Float64(x - z) * Float64(4.0 / y));
	else
		tmp = fma(Float64(4.0 / y), x, 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, -5e+21], N[Not[LessEqual[t$95$0, 5.0]], $MachinePrecision]], N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(4.0 / y), $MachinePrecision] * x + 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) < -5e21 or 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - z}{y}} \]
   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 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.0%

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

   if -5e21 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5

   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. div-add N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-lft-identity N/A

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

      13. associate-*l/ N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -5 \cdot 10^{+21} \lor \neg \left(\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 5\right):\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+16} \lor \neg \left(t\_0 \leq 5\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (or (<= t_0 -2e+16) (not (<= t_0 5.0))) (* (/ 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 <= -2e+16) || !(t_0 <= 5.0)) {
		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 <= (-2d+16)) .or. (.not. (t_0 <= 5.0d0))) 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 <= -2e+16) || !(t_0 <= 5.0)) {
		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 <= -2e+16) or not (t_0 <= 5.0):
		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 <= -2e+16) || !(t_0 <= 5.0))
		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 <= -2e+16) || ~((t_0 <= 5.0)))
		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, -2e+16], N[Not[LessEqual[t$95$0, 5.0]], $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) < -2e16 or 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - z}{y}} \]
   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 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.5%

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

   if -2e16 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5

   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 inf

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

   5. Applied rewrites 98.0%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -2 \cdot 10^{+16} \lor \neg \left(\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 5\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+167} \lor \neg \left(z \leq 4.5 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.3e+167) (not (<= z 4.5e+107)))
   (* (/ z y) -4.0)
   (fma (/ 4.0 y) x 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.3e+167) || !(z <= 4.5e+107)) {
		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 <= -1.3e+167) || !(z <= 4.5e+107))
		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, -1.3e+167], N[Not[LessEqual[z, 4.5e+107]], $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 < -1.3000000000000001e167 or 4.5e107 < z

   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 94.1

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.3%

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

   if -1.3000000000000001e167 < z < 4.5e107

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. div-add N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-lft-identity N/A

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

      13. associate-*l/ N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 86.0

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

   5. Applied rewrites 86.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+167} \lor \neg \left(z \leq 4.5 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.3% 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 99.6%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 34.8%

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

Reproduce

herbie shell --seed 2024326 
(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)))