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

Percentage Accurate: 97.8% → 100.0%

Time: 7.6s

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: 97.8% 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 98.8%

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

   1. *-commutative 98.8%

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

   2. distribute-lft-out-- 98.8%

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

   3. *-rgt-identity 98.8%

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

   4. cancel-sign-sub-inv 98.8%

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

   5. +-commutative 98.8%

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

   6. +-commutative 98.8%

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

   7. associate-+l+ 98.8%

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

   8. +-commutative 98.8%

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

   9. *-commutative 98.8%

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

   10. distribute-rgt-out 100.0%

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

   11. fma-def 100.0%

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

   12. +-commutative 100.0%

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

   13. 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: 60.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{+127}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{+32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-58}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-79}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+31}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+134}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- x))))
   (if (<= x -2e+127)
     (* x z)
     (if (<= x -1.2e+32)
       t_0
       (if (<= x -9e-58)
         (* x z)
         (if (<= x 8.5e-79)
           y
           (if (<= x 6.8e+31) (* x z) (if (<= x 1.35e+134) t_0 (* x z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * -x;
	double tmp;
	if (x <= -2e+127) {
		tmp = x * z;
	} else if (x <= -1.2e+32) {
		tmp = t_0;
	} else if (x <= -9e-58) {
		tmp = x * z;
	} else if (x <= 8.5e-79) {
		tmp = y;
	} else if (x <= 6.8e+31) {
		tmp = x * z;
	} else if (x <= 1.35e+134) {
		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 = y * -x
    if (x <= (-2d+127)) then
        tmp = x * z
    else if (x <= (-1.2d+32)) then
        tmp = t_0
    else if (x <= (-9d-58)) then
        tmp = x * z
    else if (x <= 8.5d-79) then
        tmp = y
    else if (x <= 6.8d+31) then
        tmp = x * z
    else if (x <= 1.35d+134) 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 = y * -x;
	double tmp;
	if (x <= -2e+127) {
		tmp = x * z;
	} else if (x <= -1.2e+32) {
		tmp = t_0;
	} else if (x <= -9e-58) {
		tmp = x * z;
	} else if (x <= 8.5e-79) {
		tmp = y;
	} else if (x <= 6.8e+31) {
		tmp = x * z;
	} else if (x <= 1.35e+134) {
		tmp = t_0;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * -x
	tmp = 0
	if x <= -2e+127:
		tmp = x * z
	elif x <= -1.2e+32:
		tmp = t_0
	elif x <= -9e-58:
		tmp = x * z
	elif x <= 8.5e-79:
		tmp = y
	elif x <= 6.8e+31:
		tmp = x * z
	elif x <= 1.35e+134:
		tmp = t_0
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(-x))
	tmp = 0.0
	if (x <= -2e+127)
		tmp = Float64(x * z);
	elseif (x <= -1.2e+32)
		tmp = t_0;
	elseif (x <= -9e-58)
		tmp = Float64(x * z);
	elseif (x <= 8.5e-79)
		tmp = y;
	elseif (x <= 6.8e+31)
		tmp = Float64(x * z);
	elseif (x <= 1.35e+134)
		tmp = t_0;
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * -x;
	tmp = 0.0;
	if (x <= -2e+127)
		tmp = x * z;
	elseif (x <= -1.2e+32)
		tmp = t_0;
	elseif (x <= -9e-58)
		tmp = x * z;
	elseif (x <= 8.5e-79)
		tmp = y;
	elseif (x <= 6.8e+31)
		tmp = x * z;
	elseif (x <= 1.35e+134)
		tmp = t_0;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * (-x)), $MachinePrecision]}, If[LessEqual[x, -2e+127], N[(x * z), $MachinePrecision], If[LessEqual[x, -1.2e+32], t$95$0, If[LessEqual[x, -9e-58], N[(x * z), $MachinePrecision], If[LessEqual[x, 8.5e-79], y, If[LessEqual[x, 6.8e+31], N[(x * z), $MachinePrecision], If[LessEqual[x, 1.35e+134], t$95$0, N[(x * z), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999991e127 or -1.19999999999999996e32 < x < -9.0000000000000006e-58 or 8.50000000000000029e-79 < x < 6.7999999999999996e31 or 1.35e134 < x

   1. Initial program 97.3%

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

   2. Taylor expanded in y around 0 66.6%

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

   if -1.99999999999999991e127 < x < -1.19999999999999996e32 or 6.7999999999999996e31 < x < 1.35e134

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in z around 0 79.9%

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

      1. mul-1-neg 79.9%

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

      2. distribute-lft-neg-out 79.9%

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

      3. *-commutative 79.9%

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

   7. Simplified 79.9%

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

   if -9.0000000000000006e-58 < x < 8.50000000000000029e-79

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 79.7%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+127}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{+32}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-58}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-79}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+31}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 3: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-57} \lor \neg \left(x \leq 2.1 \cdot 10^{-68}\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8e-57) (not (<= x 2.1e-68))) (* x (- z y)) y))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8e-57) || !(x <= 2.1e-68)) {
		tmp = x * (z - y);
	} else {
		tmp = 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 <= (-8d-57)) .or. (.not. (x <= 2.1d-68))) then
        tmp = x * (z - y)
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8e-57) || !(x <= 2.1e-68)) {
		tmp = x * (z - y);
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8e-57) or not (x <= 2.1e-68):
		tmp = x * (z - y)
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8e-57) || !(x <= 2.1e-68))
		tmp = Float64(x * Float64(z - y));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8e-57) || ~((x <= 2.1e-68)))
		tmp = x * (z - y);
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8e-57], N[Not[LessEqual[x, 2.1e-68]], $MachinePrecision]], N[(x * N[(z - y), $MachinePrecision]), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if x < -7.99999999999999964e-57 or 2.10000000000000008e-68 < x

   1. Initial program 97.8%

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

   2. Taylor expanded in x around inf 91.4%

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

      1. mul-1-neg 91.4%

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

      2. unsub-neg 91.4%

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

   4. Simplified 91.4%

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

   if -7.99999999999999964e-57 < x < 2.10000000000000008e-68

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 78.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-57} \lor \neg \left(x \leq 2.1 \cdot 10^{-68}\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 4: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -105 \lor \neg \left(x \leq 8 \cdot 10^{-68}\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -105.0) (not (<= x 8e-68))) (* x (- z y)) (* y (- 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -105.0) || !(x <= 8e-68)) {
		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 <= (-105.0d0)) .or. (.not. (x <= 8d-68))) 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 <= -105.0) || !(x <= 8e-68)) {
		tmp = x * (z - y);
	} else {
		tmp = y * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -105.0) or not (x <= 8e-68):
		tmp = x * (z - y)
	else:
		tmp = y * (1.0 - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -105.0) || !(x <= 8e-68))
		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 <= -105.0) || ~((x <= 8e-68)))
		tmp = x * (z - y);
	else
		tmp = y * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -105.0], N[Not[LessEqual[x, 8e-68]], $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 < -105 or 8.00000000000000053e-68 < x

   1. Initial program 97.6%

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

   2. Taylor expanded in x around inf 95.7%

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

      1. mul-1-neg 95.7%

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

      2. unsub-neg 95.7%

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

   4. Simplified 95.7%

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

   if -105 < x < 8.00000000000000053e-68

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 75.9%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -105 \lor \neg \left(x \leq 8 \cdot 10^{-68}\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.68\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.68))) (* x (- z y)) (+ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.68)) {
		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 <= 0.68d0))) 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 <= 0.68)) {
		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 <= 0.68):
		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 <= 0.68))
		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 <= 0.68)))
		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, 0.68]], $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 0.680000000000000049 < x

   1. Initial program 97.3%

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

   2. Taylor expanded in x around inf 99.0%

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

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

   4. Simplified 99.0%

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

   if -1 < x < 0.680000000000000049

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto y + x \cdot \left(\color{blue}{\left(0 - y\right)} + z\right) \]

      4. associate-+l- 100.0%

         \[\leadsto y + x \cdot \color{blue}{\left(0 - \left(y - z\right)\right)} \]

      5. unsub-neg 100.0%

         \[\leadsto y + x \cdot \left(0 - \color{blue}{\left(y + \left(-z\right)\right)}\right) \]

      6. mul-1-neg 100.0%

         \[\leadsto y + x \cdot \left(0 - \left(y + \color{blue}{-1 \cdot z}\right)\right) \]

      7. neg-sub0 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. neg-sub0 100.0%

          \[\leadsto y + x \cdot \color{blue}{\left(0 - \left(y + -1 \cdot z\right)\right)} \]

      11. mul-1-neg 100.0%

          \[\leadsto y + x \cdot \left(0 - \left(y + \color{blue}{\left(-z\right)}\right)\right) \]

      12. unsub-neg 100.0%

          \[\leadsto y + x \cdot \left(0 - \color{blue}{\left(y - z\right)}\right) \]

      13. associate-+l- 100.0%

          \[\leadsto y + x \cdot \color{blue}{\left(\left(0 - y\right) + z\right)} \]

      14. neg-sub0 100.0%

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

      15. mul-1-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. neg-mul-1 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-+r+ 100.0%

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

      6. +-commutative 100.0%

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

      7. cancel-sign-sub-inv 100.0%

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

      8. distribute-lft-out-- 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in z around inf 99.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.68\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot z\\ \end{array} \]

Alternative 6: 60.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-57}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-78}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8e-57) (* x z) (if (<= x 1.3e-78) y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8e-57) {
		tmp = x * z;
	} else if (x <= 1.3e-78) {
		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 <= (-8d-57)) then
        tmp = x * z
    else if (x <= 1.3d-78) 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 <= -8e-57) {
		tmp = x * z;
	} else if (x <= 1.3e-78) {
		tmp = y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -8e-57:
		tmp = x * z
	elif x <= 1.3e-78:
		tmp = y
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8e-57)
		tmp = Float64(x * z);
	elseif (x <= 1.3e-78)
		tmp = y;
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -8e-57)
		tmp = x * z;
	elseif (x <= 1.3e-78)
		tmp = y;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8e-57], N[(x * z), $MachinePrecision], If[LessEqual[x, 1.3e-78], y, N[(x * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.99999999999999964e-57 or 1.3000000000000001e-78 < x

   1. Initial program 97.9%

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

   2. Taylor expanded in y around 0 57.8%

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

   if -7.99999999999999964e-57 < x < 1.3000000000000001e-78

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 79.7%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-57}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-78}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

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 98.8%

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

2. Taylor expanded in x around 0 100.0%

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

   1. +-commutative 100.0%

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

   2. mul-1-neg 100.0%

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

   3. neg-sub0 100.0%

      \[\leadsto y + x \cdot \left(\color{blue}{\left(0 - y\right)} + z\right) \]

   4. associate-+l- 100.0%

      \[\leadsto y + x \cdot \color{blue}{\left(0 - \left(y - z\right)\right)} \]

   5. unsub-neg 100.0%

      \[\leadsto y + x \cdot \left(0 - \color{blue}{\left(y + \left(-z\right)\right)}\right) \]

   6. mul-1-neg 100.0%

      \[\leadsto y + x \cdot \left(0 - \left(y + \color{blue}{-1 \cdot z}\right)\right) \]

   7. neg-sub0 100.0%

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

   8. distribute-rgt-neg-in 100.0%

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

   9. distribute-rgt-neg-in 100.0%

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

   10. neg-sub0 100.0%

       \[\leadsto y + x \cdot \color{blue}{\left(0 - \left(y + -1 \cdot z\right)\right)} \]

   11. mul-1-neg 100.0%

       \[\leadsto y + x \cdot \left(0 - \left(y + \color{blue}{\left(-z\right)}\right)\right) \]

   12. unsub-neg 100.0%

       \[\leadsto y + x \cdot \left(0 - \color{blue}{\left(y - z\right)}\right) \]

   13. associate-+l- 100.0%

       \[\leadsto y + x \cdot \color{blue}{\left(\left(0 - y\right) + z\right)} \]

   14. neg-sub0 100.0%

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

   15. mul-1-neg 100.0%

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

   16. +-commutative 100.0%

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

   17. mul-1-neg 100.0%

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

4. Simplified 100.0%

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

5. Taylor expanded in y around 0 98.8%

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

   1. +-commutative 98.8%

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

   2. neg-mul-1 98.8%

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

   3. distribute-rgt-in 98.8%

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

   4. *-lft-identity 98.8%

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

   5. associate-+r+ 98.8%

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

   6. +-commutative 98.8%

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

   7. cancel-sign-sub-inv 98.8%

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

   8. distribute-lft-out-- 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 8: 36.5% 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 98.8%

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

2. Taylor expanded in x around 0 40.3%

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

3. Final simplification 40.3%

   \[\leadsto y \]

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 2023287 
(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)))