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

Percentage Accurate: 97.7% → 100.0%

Time: 4.5s

Alternatives: 8

Speedup: 1.3×

• 21.3% 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: 97.7% 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: 60.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+204}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{+142}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-39}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+247}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- x))))
   (if (<= x -1.35e+204)
     t_0
     (if (<= x -2.3e+142)
       (* x z)
       (if (<= x -1.65e+91)
         t_0
         (if (<= x -5.4e-39)
           (* x z)
           (if (<= x 1.5) y (if (<= x 2.45e+247) t_0 (* x z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * -x;
	double tmp;
	if (x <= -1.35e+204) {
		tmp = t_0;
	} else if (x <= -2.3e+142) {
		tmp = x * z;
	} else if (x <= -1.65e+91) {
		tmp = t_0;
	} else if (x <= -5.4e-39) {
		tmp = x * z;
	} else if (x <= 1.5) {
		tmp = y;
	} else if (x <= 2.45e+247) {
		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 <= (-1.35d+204)) then
        tmp = t_0
    else if (x <= (-2.3d+142)) then
        tmp = x * z
    else if (x <= (-1.65d+91)) then
        tmp = t_0
    else if (x <= (-5.4d-39)) then
        tmp = x * z
    else if (x <= 1.5d0) then
        tmp = y
    else if (x <= 2.45d+247) 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 <= -1.35e+204) {
		tmp = t_0;
	} else if (x <= -2.3e+142) {
		tmp = x * z;
	} else if (x <= -1.65e+91) {
		tmp = t_0;
	} else if (x <= -5.4e-39) {
		tmp = x * z;
	} else if (x <= 1.5) {
		tmp = y;
	} else if (x <= 2.45e+247) {
		tmp = t_0;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * -x
	tmp = 0
	if x <= -1.35e+204:
		tmp = t_0
	elif x <= -2.3e+142:
		tmp = x * z
	elif x <= -1.65e+91:
		tmp = t_0
	elif x <= -5.4e-39:
		tmp = x * z
	elif x <= 1.5:
		tmp = y
	elif x <= 2.45e+247:
		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 <= -1.35e+204)
		tmp = t_0;
	elseif (x <= -2.3e+142)
		tmp = Float64(x * z);
	elseif (x <= -1.65e+91)
		tmp = t_0;
	elseif (x <= -5.4e-39)
		tmp = Float64(x * z);
	elseif (x <= 1.5)
		tmp = y;
	elseif (x <= 2.45e+247)
		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 <= -1.35e+204)
		tmp = t_0;
	elseif (x <= -2.3e+142)
		tmp = x * z;
	elseif (x <= -1.65e+91)
		tmp = t_0;
	elseif (x <= -5.4e-39)
		tmp = x * z;
	elseif (x <= 1.5)
		tmp = y;
	elseif (x <= 2.45e+247)
		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, -1.35e+204], t$95$0, If[LessEqual[x, -2.3e+142], N[(x * z), $MachinePrecision], If[LessEqual[x, -1.65e+91], t$95$0, If[LessEqual[x, -5.4e-39], N[(x * z), $MachinePrecision], If[LessEqual[x, 1.5], y, If[LessEqual[x, 2.45e+247], t$95$0, N[(x * z), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35e204 or -2.30000000000000002e142 < x < -1.65000000000000009e91 or 1.5 < x < 2.4499999999999999e247

   1. Initial program 97.5%

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

   2. Taylor expanded in x around inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. unsub-neg 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in z around 0 69.4%

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

      1. mul-1-neg 69.4%

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

      2. distribute-lft-neg-out 69.4%

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

      3. *-commutative 69.4%

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

   7. Simplified 69.4%

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

   if -1.35e204 < x < -2.30000000000000002e142 or -1.65000000000000009e91 < x < -5.4000000000000001e-39 or 2.4499999999999999e247 < x

   1. Initial program 98.4%

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

   2. Taylor expanded in y around 0 65.6%

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

   if -5.4000000000000001e-39 < x < 1.5

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 80.2%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+204}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{+142}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{+91}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-39}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+247}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 3: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-40} \lor \neg \left(x \leq 0.043\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.25e-40) (not (<= x 0.043))) (* x (- z y)) y))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.25e-40) || !(x <= 0.043)) {
		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 <= (-1.25d-40)) .or. (.not. (x <= 0.043d0))) 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 <= -1.25e-40) || !(x <= 0.043)) {
		tmp = x * (z - y);
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.25e-40) or not (x <= 0.043):
		tmp = x * (z - y)
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.25e-40) || !(x <= 0.043))
		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 <= -1.25e-40) || ~((x <= 0.043)))
		tmp = x * (z - y);
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.25e-40], N[Not[LessEqual[x, 0.043]], $MachinePrecision]], N[(x * N[(z - y), $MachinePrecision]), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if x < -1.24999999999999991e-40 or 0.042999999999999997 < x

   1. Initial program 97.9%

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

   2. Taylor expanded in x around inf 97.2%

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

      1. mul-1-neg 97.2%

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

      2. unsub-neg 97.2%

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

   4. Simplified 97.2%

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

   if -1.24999999999999991e-40 < x < 0.042999999999999997

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 81.6%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-40} \lor \neg \left(x \leq 0.043\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 4: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-39} \lor \neg \left(x \leq 0.92\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.9e-39) (not (<= x 0.92))) (* x (- z y)) (* y (- 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.9e-39) || !(x <= 0.92)) {
		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 <= (-1.9d-39)) .or. (.not. (x <= 0.92d0))) 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 <= -1.9e-39) || !(x <= 0.92)) {
		tmp = x * (z - y);
	} else {
		tmp = y * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.9e-39) or not (x <= 0.92):
		tmp = x * (z - y)
	else:
		tmp = y * (1.0 - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.9e-39) || !(x <= 0.92))
		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 <= -1.9e-39) || ~((x <= 0.92)))
		tmp = x * (z - y);
	else
		tmp = y * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.9e-39], N[Not[LessEqual[x, 0.92]], $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 < -1.9000000000000001e-39 or 0.92000000000000004 < x

   1. Initial program 97.9%

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

   2. Taylor expanded in x around inf 97.2%

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

      1. mul-1-neg 97.2%

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

      2. unsub-neg 97.2%

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

   4. Simplified 97.2%

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

   if -1.9000000000000001e-39 < x < 0.92000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 84.5%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-39} \lor \neg \left(x \leq 0.92\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.92\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.92))) (* x (- z y)) (+ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.92)) {
		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 <= 0.92d0))) 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 <= 0.92)) {
		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 <= 0.92):
		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 <= 0.92))
		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 <= 0.92)))
		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, 0.92]], $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 0.92000000000000004 < x

   1. Initial program 97.8%

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

   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 < 0.92000000000000004

   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. 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. neg-mul-1 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. mul-1-neg 100.0%

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

      6. *-lft-identity 100.0%

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

      7. associate-+r+ 100.0%

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

      8. mul-1-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. 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 97.3%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.92\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y + x \cdot z\\ \end{array} \]

Alternative 6: 62.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-39} \lor \neg \left(x \leq 0.044\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6.5e-39) (not (<= x 0.044))) (* x z) y))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.5e-39) || !(x <= 0.044)) {
		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 <= (-6.5d-39)) .or. (.not. (x <= 0.044d0))) 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 <= -6.5e-39) || !(x <= 0.044)) {
		tmp = x * z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -6.5e-39) or not (x <= 0.044):
		tmp = x * z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6.5e-39) || !(x <= 0.044))
		tmp = Float64(x * z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -6.5e-39) || ~((x <= 0.044)))
		tmp = x * z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6.5e-39], N[Not[LessEqual[x, 0.044]], $MachinePrecision]], N[(x * z), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if x < -6.50000000000000027e-39 or 0.043999999999999997 < x

   1. Initial program 97.9%

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

   2. Taylor expanded in y around 0 50.6%

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

   if -6.50000000000000027e-39 < x < 0.043999999999999997

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 81.6%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-39} \lor \neg \left(x \leq 0.044\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

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 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. 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. neg-mul-1 98.8%

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

   2. +-commutative 98.8%

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

   3. distribute-rgt-in 98.8%

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

   4. distribute-lft-neg-out 98.8%

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

   5. mul-1-neg 98.8%

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

   6. *-lft-identity 98.8%

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

   7. associate-+r+ 98.8%

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

   8. mul-1-neg 98.8%

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

   9. sub-neg 98.8%

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

   10. +-commutative 98.8%

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

   11. 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: 36.1% 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 37.9%

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

3. Final simplification 37.9%

   \[\leadsto y \]

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 2023311 
(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)))