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

Percentage Accurate: 99.8% → 99.8%

Time: 3.4s

Alternatives: 5

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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]

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 85.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+97} \lor \neg \left(x \leq 1.8 \cdot 10^{+92}\right):\\ \;\;\;\;4 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z \cdot -4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.4e+97) (not (<= x 1.8e+92)))
   (+ 4.0 (/ (* 4.0 x) y))
   (+ 4.0 (/ (* z -4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.4e+97) || !(x <= 1.8e+92)) {
		tmp = 4.0 + ((4.0 * x) / y);
	} else {
		tmp = 4.0 + ((z * -4.0) / y);
	}
	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) :: tmp
    if ((x <= (-5.4d+97)) .or. (.not. (x <= 1.8d+92))) then
        tmp = 4.0d0 + ((4.0d0 * x) / y)
    else
        tmp = 4.0d0 + ((z * (-4.0d0)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.4e+97) || !(x <= 1.8e+92)) {
		tmp = 4.0 + ((4.0 * x) / y);
	} else {
		tmp = 4.0 + ((z * -4.0) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.4e+97) or not (x <= 1.8e+92):
		tmp = 4.0 + ((4.0 * x) / y)
	else:
		tmp = 4.0 + ((z * -4.0) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.4e+97) || !(x <= 1.8e+92))
		tmp = Float64(4.0 + Float64(Float64(4.0 * x) / y));
	else
		tmp = Float64(4.0 + Float64(Float64(z * -4.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5.4e+97) || ~((x <= 1.8e+92)))
		tmp = 4.0 + ((4.0 * x) / y);
	else
		tmp = 4.0 + ((z * -4.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.4e+97], N[Not[LessEqual[x, 1.8e+92]], $MachinePrecision]], N[(4.0 + N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.39999999999999987e97 or 1.8e92 < x

   1. Initial program 99.9%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--l+ 99.8%

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

      4. distribute-lft-in 99.8%

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

      5. associate-+r+ 99.8%

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

      6. *-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. fma-def 99.8%

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

      9. associate-*r* 99.8%

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

      10. associate-*l/ 99.8%

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

      11. associate-/l* 99.8%

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

      12. *-inverses 99.8%

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

      13. metadata-eval 99.8%

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

      14. metadata-eval 99.8%

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

      15. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 89.0%

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

      1. associate-*r/ 89.0%

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

   6. Simplified 89.0%

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

   if -5.39999999999999987e97 < x < 1.8e92

   1. Initial program 99.9%

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

      1. associate-*l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate--l+ 99.7%

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

      4. distribute-lft-in 99.8%

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

      5. associate-+r+ 99.8%

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

      6. *-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. fma-def 99.8%

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

      9. associate-*r* 99.7%

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

      10. associate-*l/ 99.8%

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

      11. associate-/l* 99.8%

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

      12. *-inverses 99.8%

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

      13. metadata-eval 99.8%

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

      14. metadata-eval 99.8%

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

      15. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 92.9%

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

      1. associate-*r/ 92.9%

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

   6. Simplified 92.9%

      \[\leadsto \color{blue}{4 + \frac{-4 \cdot z}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+97} \lor \neg \left(x \leq 1.8 \cdot 10^{+92}\right):\\ \;\;\;\;4 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z \cdot -4}{y}\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-*l/ 99.7%

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

   2. +-commutative 99.7%

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

   3. associate--l+ 99.7%

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

   4. distribute-lft-in 99.8%

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

   5. associate-+r+ 99.8%

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

   6. *-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. fma-def 99.8%

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

   9. associate-*r* 99.7%

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

   10. associate-*l/ 99.8%

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

   11. associate-/l* 99.8%

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

   12. *-inverses 99.8%

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

   13. metadata-eval 99.8%

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

   14. metadata-eval 99.8%

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

   15. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. fma-udef 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 68.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 4 + \frac{z \cdot -4}{y} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-*l/ 99.7%

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

   2. +-commutative 99.7%

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

   3. associate--l+ 99.7%

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

   4. distribute-lft-in 99.8%

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

   5. associate-+r+ 99.8%

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

   6. *-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. fma-def 99.8%

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

   9. associate-*r* 99.7%

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

   10. associate-*l/ 99.8%

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

   11. associate-/l* 99.8%

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

   12. *-inverses 99.8%

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

   13. metadata-eval 99.8%

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

   14. metadata-eval 99.8%

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

   15. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 71.1%

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

   1. associate-*r/ 71.1%

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

6. Simplified 71.1%

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

7. Final simplification 71.1%

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

Alternative 5: 34.6% accurate, 13.0× 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. Step-by-step derivation

   1. associate-*l/ 99.7%

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

   2. +-commutative 99.7%

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

   3. associate--l+ 99.7%

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

   4. distribute-lft-in 99.8%

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

   5. associate-+r+ 99.8%

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

   6. *-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. fma-def 99.8%

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

   9. associate-*r* 99.7%

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

   10. associate-*l/ 99.8%

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

   11. associate-/l* 99.8%

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

   12. *-inverses 99.8%

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

   13. metadata-eval 99.8%

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

   14. metadata-eval 99.8%

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

   15. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in y around inf 33.5%

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

5. Final simplification 33.5%

   \[\leadsto 4 \]

Reproduce

herbie shell --seed 2023199 
(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)))