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

Percentage Accurate: 97.6% → 100.0%

Time: 3.5s

Alternatives: 8

Speedup: 1.3×

• 21.0% 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 99.6%

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

   1. *-commutative 99.6%

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

   2. distribute-lft-out-- 99.6%

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

   3. *-rgt-identity 99.6%

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

   4. cancel-sign-sub-inv 99.6%

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

   5. +-commutative 99.6%

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

   6. +-commutative 99.6%

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

   7. associate-+l+ 99.6%

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

   8. +-commutative 99.6%

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

   9. *-commutative 99.6%

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

   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: 60.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-y\right)\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+207}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -6.3 \cdot 10^{+165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{+123}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-63}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- y))))
   (if (<= x -1.65e+207)
     (* x z)
     (if (<= x -6.3e+165)
       t_0
       (if (<= x -4.8e+123)
         (* x z)
         (if (<= x -1.0) t_0 (if (<= x 2.6e-63) y (* x z))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * -y;
	double tmp;
	if (x <= -1.65e+207) {
		tmp = x * z;
	} else if (x <= -6.3e+165) {
		tmp = t_0;
	} else if (x <= -4.8e+123) {
		tmp = x * z;
	} else if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 2.6e-63) {
		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) :: t_0
    real(8) :: tmp
    t_0 = x * -y
    if (x <= (-1.65d+207)) then
        tmp = x * z
    else if (x <= (-6.3d+165)) then
        tmp = t_0
    else if (x <= (-4.8d+123)) then
        tmp = x * z
    else if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 2.6d-63) 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 t_0 = x * -y;
	double tmp;
	if (x <= -1.65e+207) {
		tmp = x * z;
	} else if (x <= -6.3e+165) {
		tmp = t_0;
	} else if (x <= -4.8e+123) {
		tmp = x * z;
	} else if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 2.6e-63) {
		tmp = y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -y
	tmp = 0
	if x <= -1.65e+207:
		tmp = x * z
	elif x <= -6.3e+165:
		tmp = t_0
	elif x <= -4.8e+123:
		tmp = x * z
	elif x <= -1.0:
		tmp = t_0
	elif x <= 2.6e-63:
		tmp = y
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-y))
	tmp = 0.0
	if (x <= -1.65e+207)
		tmp = Float64(x * z);
	elseif (x <= -6.3e+165)
		tmp = t_0;
	elseif (x <= -4.8e+123)
		tmp = Float64(x * z);
	elseif (x <= -1.0)
		tmp = t_0;
	elseif (x <= 2.6e-63)
		tmp = y;
	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 <= -1.65e+207)
		tmp = x * z;
	elseif (x <= -6.3e+165)
		tmp = t_0;
	elseif (x <= -4.8e+123)
		tmp = x * z;
	elseif (x <= -1.0)
		tmp = t_0;
	elseif (x <= 2.6e-63)
		tmp = y;
	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, -1.65e+207], N[(x * z), $MachinePrecision], If[LessEqual[x, -6.3e+165], t$95$0, If[LessEqual[x, -4.8e+123], N[(x * z), $MachinePrecision], If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 2.6e-63], y, N[(x * z), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.65e207 or -6.3000000000000002e165 < x < -4.79999999999999978e123 or 2.6000000000000001e-63 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 67.5%

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

   if -1.65e207 < x < -6.3000000000000002e165 or -4.79999999999999978e123 < x < -1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. sub-neg 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in z around 0 66.5%

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

      1. mul-1-neg 66.5%

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

      2. distribute-rgt-neg-out 66.5%

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

   7. Simplified 66.5%

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

   if -1 < x < 2.6000000000000001e-63

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 73.7%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+207}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -6.3 \cdot 10^{+165}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{+123}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-63}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 3: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.028 \lor \neg \left(x \leq 6.8 \cdot 10^{-49}\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -0.028) (not (<= x 6.8e-49))) (* x (- z y)) y))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.028) || !(x <= 6.8e-49)) {
		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 <= (-0.028d0)) .or. (.not. (x <= 6.8d-49))) 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 <= -0.028) || !(x <= 6.8e-49)) {
		tmp = x * (z - y);
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -0.028) or not (x <= 6.8e-49):
		tmp = x * (z - y)
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -0.028) || !(x <= 6.8e-49))
		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 <= -0.028) || ~((x <= 6.8e-49)))
		tmp = x * (z - y);
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -0.028], N[Not[LessEqual[x, 6.8e-49]], $MachinePrecision]], N[(x * N[(z - y), $MachinePrecision]), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if x < -0.0280000000000000006 or 6.8000000000000001e-49 < x

   1. Initial program 99.2%

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

   2. Taylor expanded in x around inf 97.0%

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

      1. mul-1-neg 97.0%

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

      2. sub-neg 97.0%

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

   4. Simplified 97.0%

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

   if -0.0280000000000000006 < x < 6.8000000000000001e-49

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 73.4%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.028 \lor \neg \left(x \leq 6.8 \cdot 10^{-49}\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 4: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \lor \neg \left(x \leq 1.7 \cdot 10^{-48}\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.5) (not (<= x 1.7e-48))) (* x (- z y)) (* y (- 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.5) || !(x <= 1.7e-48)) {
		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 <= (-4.5d0)) .or. (.not. (x <= 1.7d-48))) 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 <= -4.5) || !(x <= 1.7e-48)) {
		tmp = x * (z - y);
	} else {
		tmp = y * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.5) or not (x <= 1.7e-48):
		tmp = x * (z - y)
	else:
		tmp = y * (1.0 - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.5) || !(x <= 1.7e-48))
		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 <= -4.5) || ~((x <= 1.7e-48)))
		tmp = x * (z - y);
	else
		tmp = y * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.5], N[Not[LessEqual[x, 1.7e-48]], $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 < -4.5 or 1.70000000000000014e-48 < x

   1. Initial program 99.2%

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

   2. Taylor expanded in x around inf 97.0%

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

      1. mul-1-neg 97.0%

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

      2. sub-neg 97.0%

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

   4. Simplified 97.0%

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

   if -4.5 < x < 1.70000000000000014e-48

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 75.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \lor \neg \left(x \leq 1.7 \cdot 10^{-48}\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 5: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -580 \lor \neg \left(x \leq 5.3 \cdot 10^{-12}\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -580.0) (not (<= x 5.3e-12))) (* x (- z y)) (+ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -580.0) || !(x <= 5.3e-12)) {
		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 <= (-580.0d0)) .or. (.not. (x <= 5.3d-12))) 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 <= -580.0) || !(x <= 5.3e-12)) {
		tmp = x * (z - y);
	} else {
		tmp = y + (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -580.0) or not (x <= 5.3e-12):
		tmp = x * (z - y)
	else:
		tmp = y + (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -580.0) || !(x <= 5.3e-12))
		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 <= -580.0) || ~((x <= 5.3e-12)))
		tmp = x * (z - y);
	else
		tmp = y + (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -580.0], N[Not[LessEqual[x, 5.3e-12]], $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 < -580 or 5.29999999999999963e-12 < x

   1. Initial program 99.1%

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

   2. Taylor expanded in x around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. sub-neg 99.1%

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

   4. Simplified 99.1%

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

   if -580 < x < 5.29999999999999963e-12

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-lft-out-- 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. *-commutative 100.0%

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

      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. Step-by-step derivation

      1. fma-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in z around inf 97.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -580 \lor \neg \left(x \leq 5.3 \cdot 10^{-12}\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot z\\ \end{array} \]

Alternative 6: 61.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.028 \lor \neg \left(x \leq 2.4 \cdot 10^{-63}\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -0.028) (not (<= x 2.4e-63))) (* x z) y))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.028) || !(x <= 2.4e-63)) {
		tmp = x * z;
	} 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 <= (-0.028d0)) .or. (.not. (x <= 2.4d-63))) then
        tmp = x * z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.028) || !(x <= 2.4e-63)) {
		tmp = x * z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -0.028) or not (x <= 2.4e-63):
		tmp = x * z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -0.028) || !(x <= 2.4e-63))
		tmp = Float64(x * z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -0.028) || ~((x <= 2.4e-63)))
		tmp = x * z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -0.028], N[Not[LessEqual[x, 2.4e-63]], $MachinePrecision]], N[(x * z), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if x < -0.0280000000000000006 or 2.4000000000000001e-63 < x

   1. Initial program 99.2%

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

   2. Taylor expanded in y around 0 59.7%

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

   if -0.0280000000000000006 < x < 2.4000000000000001e-63

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 73.7%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.028 \lor \neg \left(x \leq 2.4 \cdot 10^{-63}\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

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 99.6%

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

   1. *-commutative 99.6%

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

   2. distribute-lft-out-- 99.6%

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

   3. *-rgt-identity 99.6%

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

   4. cancel-sign-sub-inv 99.6%

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

   5. +-commutative 99.6%

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

   6. +-commutative 99.6%

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

   7. associate-+l+ 99.6%

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

   8. +-commutative 99.6%

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

   9. *-commutative 99.6%

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

   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. Step-by-step derivation

   1. fma-udef 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 8: 36.6% 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 99.6%

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

2. Taylor expanded in x around 0 38.6%

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

3. Final simplification 38.6%

   \[\leadsto y \]

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 2023320 
(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)))