Diagrams.Color.HSV:lerp from diagrams-contrib-1.3.0.5

Percentage Accurate: 98.0% → 100.0%

Time: 4.4s

Alternatives: 8

Speedup: 1.3×

• 20.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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 - x) * y) + (x * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - x) * y) + (x * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. sub-neg 98.4%

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

   2. +-commutative 98.4%

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

   3. distribute-rgt1-in 98.4%

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

   4. associate-+l+ 98.4%

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

   5. +-commutative 98.4%

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

   6. *-commutative 98.4%

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

   7. neg-mul-1 98.4%

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

   8. associate-*r* 98.4%

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

   9. *-commutative 98.4%

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

   10. distribute-rgt-out 100.0%

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

   11. fma-def 100.0%

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

   12. +-commutative 100.0%

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

   13. *-commutative 100.0%

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

   14. neg-mul-1 100.0%

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

   15. unsub-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-14} \lor \neg \left(y \leq -2.25 \cdot 10^{-91}\right) \land \left(y \leq -2.1 \cdot 10^{-152} \lor \neg \left(y \leq 2.75 \cdot 10^{-73}\right)\right):\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.2e-14)
         (and (not (<= y -2.25e-91))
              (or (<= y -2.1e-152) (not (<= y 2.75e-73)))))
   (* y (- 1.0 x))
   (* x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.2e-14) || (!(y <= -2.25e-91) && ((y <= -2.1e-152) || !(y <= 2.75e-73)))) {
		tmp = y * (1.0 - x);
	} else {
		tmp = x * z;
	}
	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 ((y <= (-4.2d-14)) .or. (.not. (y <= (-2.25d-91))) .and. (y <= (-2.1d-152)) .or. (.not. (y <= 2.75d-73))) then
        tmp = y * (1.0d0 - x)
    else
        tmp = x * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.2e-14) || (!(y <= -2.25e-91) && ((y <= -2.1e-152) || !(y <= 2.75e-73)))) {
		tmp = y * (1.0 - x);
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.2e-14) or (not (y <= -2.25e-91) and ((y <= -2.1e-152) or not (y <= 2.75e-73))):
		tmp = y * (1.0 - x)
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.2e-14) || (!(y <= -2.25e-91) && ((y <= -2.1e-152) || !(y <= 2.75e-73))))
		tmp = Float64(y * Float64(1.0 - x));
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4.2e-14) || (~((y <= -2.25e-91)) && ((y <= -2.1e-152) || ~((y <= 2.75e-73)))))
		tmp = y * (1.0 - x);
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.2e-14], And[N[Not[LessEqual[y, -2.25e-91]], $MachinePrecision], Or[LessEqual[y, -2.1e-152], N[Not[LessEqual[y, 2.75e-73]], $MachinePrecision]]]], N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(x * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.1999999999999998e-14 or -2.24999999999999988e-91 < y < -2.09999999999999999e-152 or 2.75000000000000003e-73 < y

   1. Initial program 97.5%

      \[\left(1 - x\right) \cdot y + x \cdot z \]

   2. Taylor expanded in y around inf 81.9%

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

   if -4.1999999999999998e-14 < y < -2.24999999999999988e-91 or -2.09999999999999999e-152 < y < 2.75000000000000003e-73

   1. Initial program 100.0%

      \[\left(1 - x\right) \cdot y + x \cdot z \]

   2. Taylor expanded in y around 0 75.1%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-14} \lor \neg \left(y \leq -2.25 \cdot 10^{-91}\right) \land \left(y \leq -2.1 \cdot 10^{-152} \lor \neg \left(y \leq 2.75 \cdot 10^{-73}\right)\right):\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 3: 61.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-48}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-23}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+197}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+255}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.45e-48)
   (* x z)
   (if (<= x 8.6e-23)
     y
     (if (<= x 1.5e+197) (* x z) (if (<= x 2.5e+255) (* y (- x)) (* x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45e-48) {
		tmp = x * z;
	} else if (x <= 8.6e-23) {
		tmp = y;
	} else if (x <= 1.5e+197) {
		tmp = x * z;
	} else if (x <= 2.5e+255) {
		tmp = y * -x;
	} else {
		tmp = x * z;
	}
	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 <= (-1.45d-48)) then
        tmp = x * z
    else if (x <= 8.6d-23) then
        tmp = y
    else if (x <= 1.5d+197) then
        tmp = x * z
    else if (x <= 2.5d+255) then
        tmp = y * -x
    else
        tmp = x * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45e-48) {
		tmp = x * z;
	} else if (x <= 8.6e-23) {
		tmp = y;
	} else if (x <= 1.5e+197) {
		tmp = x * z;
	} else if (x <= 2.5e+255) {
		tmp = y * -x;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.45e-48:
		tmp = x * z
	elif x <= 8.6e-23:
		tmp = y
	elif x <= 1.5e+197:
		tmp = x * z
	elif x <= 2.5e+255:
		tmp = y * -x
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.45e-48)
		tmp = Float64(x * z);
	elseif (x <= 8.6e-23)
		tmp = y;
	elseif (x <= 1.5e+197)
		tmp = Float64(x * z);
	elseif (x <= 2.5e+255)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.45e-48)
		tmp = x * z;
	elseif (x <= 8.6e-23)
		tmp = y;
	elseif (x <= 1.5e+197)
		tmp = x * z;
	elseif (x <= 2.5e+255)
		tmp = y * -x;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.45e-48], N[(x * z), $MachinePrecision], If[LessEqual[x, 8.6e-23], y, If[LessEqual[x, 1.5e+197], N[(x * z), $MachinePrecision], If[LessEqual[x, 2.5e+255], N[(y * (-x)), $MachinePrecision], N[(x * z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4500000000000001e-48 or 8.60000000000000004e-23 < x < 1.5000000000000001e197 or 2.5000000000000001e255 < x

   1. Initial program 96.8%

      \[\left(1 - x\right) \cdot y + x \cdot z \]

   2. Taylor expanded in y around 0 62.6%

      \[\leadsto \color{blue}{z \cdot x} \]

   if -1.4500000000000001e-48 < x < 8.60000000000000004e-23

   1. Initial program 100.0%

      \[\left(1 - x\right) \cdot y + x \cdot z \]

   2. Taylor expanded in x around 0 77.9%

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

   if 1.5000000000000001e197 < x < 2.5000000000000001e255

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt1-in 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. neg-mul-1 100.0%

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

      8. associate-*r* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   5. Taylor expanded in z around 0 73.9%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot x\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 73.9%

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

      2. distribute-rgt-neg-in 73.9%

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

   7. Simplified 73.9%

      \[\leadsto \color{blue}{y \cdot \left(-x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-48}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-23}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+197}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+255}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 4: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.66 \cdot 10^{-49} \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.66e-49) (not (<= x 1.5))) (* x (- z y)) (* y (- 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.66e-49) || !(x <= 1.5)) {
		tmp = x * (z - y);
	} else {
		tmp = y * (1.0 - x);
	}
	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 <= (-1.66d-49)) .or. (.not. (x <= 1.5d0))) then
        tmp = x * (z - y)
    else
        tmp = y * (1.0d0 - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.66e-49) || !(x <= 1.5)) {
		tmp = x * (z - y);
	} else {
		tmp = y * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.66e-49) or not (x <= 1.5):
		tmp = x * (z - y)
	else:
		tmp = y * (1.0 - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.66e-49) || !(x <= 1.5))
		tmp = Float64(x * Float64(z - y));
	else
		tmp = Float64(y * Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.66e-49) || ~((x <= 1.5)))
		tmp = x * (z - y);
	else
		tmp = y * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.66e-49], N[Not[LessEqual[x, 1.5]], $MachinePrecision]], N[(x * N[(z - y), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6599999999999999e-49 or 1.5 < x

   1. Initial program 97.1%

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

      1. sub-neg 97.1%

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

      2. +-commutative 97.1%

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

      3. distribute-rgt1-in 97.1%

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

      4. associate-+l+ 97.1%

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

      5. +-commutative 97.1%

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

      6. *-commutative 97.1%

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

      7. neg-mul-1 97.1%

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

      8. associate-*r* 97.1%

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

      9. *-commutative 97.1%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 96.3%

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

   if -1.6599999999999999e-49 < x < 1.5

   1. Initial program 100.0%

      \[\left(1 - x\right) \cdot y + x \cdot z \]

   2. Taylor expanded in y around inf 77.5%

      \[\leadsto \color{blue}{y \cdot \left(1 - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.66 \cdot 10^{-49} \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 5: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (* x (- z y)) (+ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = x * (z - y);
	} else {
		tmp = y + (x * z);
	}
	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 <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = x * (z - y)
    else
        tmp = y + (x * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = x * (z - y);
	} else {
		tmp = y + (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = x * (z - y)
	else:
		tmp = y + (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(x * Float64(z - y));
	else
		tmp = Float64(y + Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = x * (z - y);
	else
		tmp = y + (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(x * N[(z - y), $MachinePrecision]), $MachinePrecision], N[(y + N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 96.9%

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

      1. sub-neg 96.9%

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

      2. +-commutative 96.9%

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

      3. distribute-rgt1-in 96.9%

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

      4. associate-+l+ 96.9%

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

      5. +-commutative 96.9%

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

      6. *-commutative 96.9%

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

      7. neg-mul-1 96.9%

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

      8. associate-*r* 96.9%

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

      9. *-commutative 96.9%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.8%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt1-in 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. neg-mul-1 100.0%

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

      8. associate-*r* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

   5. Taylor expanded in z around inf 99.1%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot z\\ \end{array} \]

Alternative 6: 61.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-49}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-22}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.65e-49) (* x z) (if (<= x 2.7e-22) y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.65e-49) {
		tmp = x * z;
	} else if (x <= 2.7e-22) {
		tmp = y;
	} else {
		tmp = x * z;
	}
	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 <= (-2.65d-49)) then
        tmp = x * z
    else if (x <= 2.7d-22) then
        tmp = y
    else
        tmp = x * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.65e-49) {
		tmp = x * z;
	} else if (x <= 2.7e-22) {
		tmp = y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.65e-49:
		tmp = x * z
	elif x <= 2.7e-22:
		tmp = y
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.65e-49)
		tmp = Float64(x * z);
	elseif (x <= 2.7e-22)
		tmp = y;
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.65e-49)
		tmp = x * z;
	elseif (x <= 2.7e-22)
		tmp = y;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.65e-49], N[(x * z), $MachinePrecision], If[LessEqual[x, 2.7e-22], y, N[(x * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6500000000000001e-49 or 2.7000000000000002e-22 < x

   1. Initial program 97.2%

      \[\left(1 - x\right) \cdot y + x \cdot z \]

   2. Taylor expanded in y around 0 59.6%

      \[\leadsto \color{blue}{z \cdot x} \]

   if -2.6500000000000001e-49 < x < 2.7000000000000002e-22

   1. Initial program 100.0%

      \[\left(1 - x\right) \cdot y + x \cdot z \]

   2. Taylor expanded in x around 0 77.9%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-49}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-22}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 7: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. sub-neg 98.4%

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

   2. +-commutative 98.4%

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

   3. distribute-rgt1-in 98.4%

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

   4. associate-+l+ 98.4%

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

   5. +-commutative 98.4%

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

   6. *-commutative 98.4%

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

   7. neg-mul-1 98.4%

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

   8. associate-*r* 98.4%

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

   9. *-commutative 98.4%

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

   10. distribute-rgt-out 100.0%

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

   11. fma-def 100.0%

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

   12. +-commutative 100.0%

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

   13. *-commutative 100.0%

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

   14. neg-mul-1 100.0%

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

   15. unsub-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

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

5. Final simplification 100.0%

   \[\leadsto y + x \cdot \left(z - y\right) \]

Alternative 8: 37.5% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 y)

C

double code(double x, double y, double z) {
	return 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 = y
end function

Java

public static double code(double x, double y, double z) {
	return y;
}

Python

def code(x, y, z):
	return y

Julia

function code(x, y, z)
	return y
end

MATLAB

function tmp = code(x, y, z)
	tmp = y;
end

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 98.4%

   \[\left(1 - x\right) \cdot y + x \cdot z \]

2. Taylor expanded in x around 0 38.1%

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

3. Final simplification 38.1%

   \[\leadsto y \]

Developer target: 100.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- y (* x (- y z))))

C

double code(double x, double y, double z) {
	return y - (x * (y - z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y - (x * (y - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023200 
(FPCore (x y z)
  :name "Diagrams.Color.HSV:lerp  from diagrams-contrib-1.3.0.5"
  :precision binary64

  :herbie-target
  (- y (* x (- y z)))

  (+ (* (- 1.0 x) y) (* x z)))