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

Percentage Accurate: 97.6% → 100.0%

Time: 5.7s

Alternatives: 7

Speedup: 1.3×

• 21.5% 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: 97.6% 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 98.0%

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

   1. sub-neg 98.0%

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

   2. +-commutative 98.0%

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

   3. distribute-rgt1-in 98.0%

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

   4. associate-+l+ 98.0%

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

   5. +-commutative 98.0%

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

   6. *-commutative 98.0%

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

   7. neg-mul-1 98.0%

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

   8. associate-*r* 98.0%

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

   9. *-commutative 98.0%

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

   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. Final simplification 100.0%

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

Alternative 2: 61.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-y\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{+45}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-12}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-123}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-72}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-32}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+182}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+206}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- y))))
   (if (<= x -1e+45)
     (* x z)
     (if (<= x -1.55e+27)
       t_0
       (if (<= x -1.5e-12)
         (* x z)
         (if (<= x 7.5e-123)
           y
           (if (<= x 7.5e-72)
             (* x z)
             (if (<= x 3.7e-32)
               y
               (if (<= x 3.1e+182)
                 (* x z)
                 (if (<= x 1.6e+206) t_0 (* x z)))))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * -y;
	double tmp;
	if (x <= -1e+45) {
		tmp = x * z;
	} else if (x <= -1.55e+27) {
		tmp = t_0;
	} else if (x <= -1.5e-12) {
		tmp = x * z;
	} else if (x <= 7.5e-123) {
		tmp = y;
	} else if (x <= 7.5e-72) {
		tmp = x * z;
	} else if (x <= 3.7e-32) {
		tmp = y;
	} else if (x <= 3.1e+182) {
		tmp = x * z;
	} else if (x <= 1.6e+206) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x * -y
    if (x <= (-1d+45)) then
        tmp = x * z
    else if (x <= (-1.55d+27)) then
        tmp = t_0
    else if (x <= (-1.5d-12)) then
        tmp = x * z
    else if (x <= 7.5d-123) then
        tmp = y
    else if (x <= 7.5d-72) then
        tmp = x * z
    else if (x <= 3.7d-32) then
        tmp = y
    else if (x <= 3.1d+182) then
        tmp = x * z
    else if (x <= 1.6d+206) then
        tmp = t_0
    else
        tmp = x * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * -y;
	double tmp;
	if (x <= -1e+45) {
		tmp = x * z;
	} else if (x <= -1.55e+27) {
		tmp = t_0;
	} else if (x <= -1.5e-12) {
		tmp = x * z;
	} else if (x <= 7.5e-123) {
		tmp = y;
	} else if (x <= 7.5e-72) {
		tmp = x * z;
	} else if (x <= 3.7e-32) {
		tmp = y;
	} else if (x <= 3.1e+182) {
		tmp = x * z;
	} else if (x <= 1.6e+206) {
		tmp = t_0;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -y
	tmp = 0
	if x <= -1e+45:
		tmp = x * z
	elif x <= -1.55e+27:
		tmp = t_0
	elif x <= -1.5e-12:
		tmp = x * z
	elif x <= 7.5e-123:
		tmp = y
	elif x <= 7.5e-72:
		tmp = x * z
	elif x <= 3.7e-32:
		tmp = y
	elif x <= 3.1e+182:
		tmp = x * z
	elif x <= 1.6e+206:
		tmp = t_0
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-y))
	tmp = 0.0
	if (x <= -1e+45)
		tmp = Float64(x * z);
	elseif (x <= -1.55e+27)
		tmp = t_0;
	elseif (x <= -1.5e-12)
		tmp = Float64(x * z);
	elseif (x <= 7.5e-123)
		tmp = y;
	elseif (x <= 7.5e-72)
		tmp = Float64(x * z);
	elseif (x <= 3.7e-32)
		tmp = y;
	elseif (x <= 3.1e+182)
		tmp = Float64(x * z);
	elseif (x <= 1.6e+206)
		tmp = t_0;
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * -y;
	tmp = 0.0;
	if (x <= -1e+45)
		tmp = x * z;
	elseif (x <= -1.55e+27)
		tmp = t_0;
	elseif (x <= -1.5e-12)
		tmp = x * z;
	elseif (x <= 7.5e-123)
		tmp = y;
	elseif (x <= 7.5e-72)
		tmp = x * z;
	elseif (x <= 3.7e-32)
		tmp = y;
	elseif (x <= 3.1e+182)
		tmp = x * z;
	elseif (x <= 1.6e+206)
		tmp = t_0;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * (-y)), $MachinePrecision]}, If[LessEqual[x, -1e+45], N[(x * z), $MachinePrecision], If[LessEqual[x, -1.55e+27], t$95$0, If[LessEqual[x, -1.5e-12], N[(x * z), $MachinePrecision], If[LessEqual[x, 7.5e-123], y, If[LessEqual[x, 7.5e-72], N[(x * z), $MachinePrecision], If[LessEqual[x, 3.7e-32], y, If[LessEqual[x, 3.1e+182], N[(x * z), $MachinePrecision], If[LessEqual[x, 1.6e+206], t$95$0, N[(x * z), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.9999999999999993e44 or -1.54999999999999998e27 < x < -1.5000000000000001e-12 or 7.50000000000000011e-123 < x < 7.5000000000000004e-72 or 3.7e-32 < x < 3.09999999999999996e182 or 1.60000000000000003e206 < x

   1. Initial program 96.7%

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

   2. Taylor expanded in y around 0 66.6%

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

   if -9.9999999999999993e44 < x < -1.54999999999999998e27 or 3.09999999999999996e182 < x < 1.60000000000000003e206

   1. Initial program 95.0%

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

      1. sub-neg 95.0%

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

      2. +-commutative 95.0%

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

      3. distribute-rgt1-in 95.0%

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

      4. associate-+l+ 95.0%

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

      5. +-commutative 95.0%

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

      6. *-commutative 95.0%

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

      7. neg-mul-1 95.0%

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

      8. associate-*r* 95.0%

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

      9. *-commutative 95.0%

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

      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 80.3%

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

      1. mul-1-neg 80.3%

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

      2. distribute-rgt-neg-in 80.3%

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

   7. Simplified 80.3%

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

   if -1.5000000000000001e-12 < x < 7.50000000000000011e-123 or 7.5000000000000004e-72 < x < 3.7e-32

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 77.0%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+45}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{+27}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-12}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-123}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-72}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-32}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+182}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+206}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 3: 84.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(z - y\right)\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-126}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-72}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-31}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z y))))
   (if (<= x -5.4e-13)
     t_0
     (if (<= x 1.45e-126)
       y
       (if (<= x 8.5e-72) (* x z) (if (<= x 1.9e-31) y t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (z - y);
	double tmp;
	if (x <= -5.4e-13) {
		tmp = t_0;
	} else if (x <= 1.45e-126) {
		tmp = y;
	} else if (x <= 8.5e-72) {
		tmp = x * z;
	} else if (x <= 1.9e-31) {
		tmp = y;
	} 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) :: tmp
    t_0 = x * (z - y)
    if (x <= (-5.4d-13)) then
        tmp = t_0
    else if (x <= 1.45d-126) then
        tmp = y
    else if (x <= 8.5d-72) then
        tmp = x * z
    else if (x <= 1.9d-31) then
        tmp = y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (z - y);
	double tmp;
	if (x <= -5.4e-13) {
		tmp = t_0;
	} else if (x <= 1.45e-126) {
		tmp = y;
	} else if (x <= 8.5e-72) {
		tmp = x * z;
	} else if (x <= 1.9e-31) {
		tmp = y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (z - y)
	tmp = 0
	if x <= -5.4e-13:
		tmp = t_0
	elif x <= 1.45e-126:
		tmp = y
	elif x <= 8.5e-72:
		tmp = x * z
	elif x <= 1.9e-31:
		tmp = y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(z - y))
	tmp = 0.0
	if (x <= -5.4e-13)
		tmp = t_0;
	elseif (x <= 1.45e-126)
		tmp = y;
	elseif (x <= 8.5e-72)
		tmp = Float64(x * z);
	elseif (x <= 1.9e-31)
		tmp = y;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (z - y);
	tmp = 0.0;
	if (x <= -5.4e-13)
		tmp = t_0;
	elseif (x <= 1.45e-126)
		tmp = y;
	elseif (x <= 8.5e-72)
		tmp = x * z;
	elseif (x <= 1.9e-31)
		tmp = y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.4e-13], t$95$0, If[LessEqual[x, 1.45e-126], y, If[LessEqual[x, 8.5e-72], N[(x * z), $MachinePrecision], If[LessEqual[x, 1.9e-31], y, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.40000000000000021e-13 or 1.9e-31 < x

   1. Initial program 96.1%

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

      1. sub-neg 96.1%

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

      2. +-commutative 96.1%

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

      3. distribute-rgt1-in 96.1%

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

      4. associate-+l+ 96.1%

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

      5. +-commutative 96.1%

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

      6. *-commutative 96.1%

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

      7. neg-mul-1 96.1%

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

      8. associate-*r* 96.1%

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

      9. *-commutative 96.1%

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

      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 98.0%

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

   if -5.40000000000000021e-13 < x < 1.44999999999999994e-126 or 8.50000000000000008e-72 < x < 1.9e-31

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 77.0%

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

   if 1.44999999999999994e-126 < x < 8.50000000000000008e-72

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 84.3%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-126}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-72}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-31}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z - y\right)\\ \end{array} \]

Alternative 4: 61.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-124}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-73}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-32}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.7e-8)
   (* x z)
   (if (<= x 3.9e-124)
     y
     (if (<= x 5.6e-73) (* x z) (if (<= x 1.75e-32) y (* x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.7e-8) {
		tmp = x * z;
	} else if (x <= 3.9e-124) {
		tmp = y;
	} else if (x <= 5.6e-73) {
		tmp = x * z;
	} else if (x <= 1.75e-32) {
		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 <= (-2.7d-8)) then
        tmp = x * z
    else if (x <= 3.9d-124) then
        tmp = y
    else if (x <= 5.6d-73) then
        tmp = x * z
    else if (x <= 1.75d-32) 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 <= -2.7e-8) {
		tmp = x * z;
	} else if (x <= 3.9e-124) {
		tmp = y;
	} else if (x <= 5.6e-73) {
		tmp = x * z;
	} else if (x <= 1.75e-32) {
		tmp = y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.7e-8:
		tmp = x * z
	elif x <= 3.9e-124:
		tmp = y
	elif x <= 5.6e-73:
		tmp = x * z
	elif x <= 1.75e-32:
		tmp = y
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.7e-8)
		tmp = Float64(x * z);
	elseif (x <= 3.9e-124)
		tmp = y;
	elseif (x <= 5.6e-73)
		tmp = Float64(x * z);
	elseif (x <= 1.75e-32)
		tmp = y;
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.7e-8)
		tmp = x * z;
	elseif (x <= 3.9e-124)
		tmp = y;
	elseif (x <= 5.6e-73)
		tmp = x * z;
	elseif (x <= 1.75e-32)
		tmp = y;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.7e-8], N[(x * z), $MachinePrecision], If[LessEqual[x, 3.9e-124], y, If[LessEqual[x, 5.6e-73], N[(x * z), $MachinePrecision], If[LessEqual[x, 1.75e-32], y, N[(x * z), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.70000000000000002e-8 or 3.89999999999999993e-124 < x < 5.60000000000000023e-73 or 1.7499999999999999e-32 < x

   1. Initial program 96.4%

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

   2. Taylor expanded in y around 0 60.4%

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

   if -2.70000000000000002e-8 < x < 3.89999999999999993e-124 or 5.60000000000000023e-73 < x < 1.7499999999999999e-32

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 77.0%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-124}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-73}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-32}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 5: 98.9% 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 95.9%

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

      1. sub-neg 95.9%

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

      2. +-commutative 95.9%

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

      3. distribute-rgt1-in 95.9%

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

      4. associate-+l+ 95.9%

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

      5. +-commutative 95.9%

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

      6. *-commutative 95.9%

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

      7. neg-mul-1 95.9%

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

      8. associate-*r* 95.9%

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

      9. *-commutative 95.9%

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

      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 99.5%

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

   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 99.2%

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

4. Final simplification 99.4%

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

Alternative 6: 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 98.0%

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

   1. sub-neg 98.0%

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

   2. +-commutative 98.0%

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

   3. distribute-rgt1-in 98.0%

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

   4. associate-+l+ 98.0%

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

   5. +-commutative 98.0%

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

   6. *-commutative 98.0%

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

   7. neg-mul-1 98.0%

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

   8. associate-*r* 98.0%

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

   9. *-commutative 98.0%

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

   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. Final simplification 100.0%

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

Alternative 7: 36.4% 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 98.0%

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

2. Taylor expanded in x around 0 37.0%

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

3. Final simplification 37.0%

   \[\leadsto y \]

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 2023199 
(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)))