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

Percentage Accurate: 97.6% → 100.0%

Time: 5.7s

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: 97.6% 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 97.6%

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

   1. *-commutative 97.6%

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

   2. distribute-lft-out-- 97.6%

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

   3. *-rgt-identity 97.6%

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

   4. cancel-sign-sub-inv 97.6%

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

   5. +-commutative 97.6%

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

   6. +-commutative 97.6%

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

   7. associate-+l+ 97.6%

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

   8. +-commutative 97.6%

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

   9. *-commutative 97.6%

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

   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: 59.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.82 \cdot 10^{+208}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{+113}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-39}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-158}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-105}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-15}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.82e+208)
   (* x z)
   (if (<= x -1.6e+113)
     (* x (- y))
     (if (<= x -1.9e-39)
       (* x z)
       (if (<= x 1.7e-158)
         y
         (if (<= x 5.7e-105) (* x z) (if (<= x 1.6e-15) y (* x z))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.82e+208) {
		tmp = x * z;
	} else if (x <= -1.6e+113) {
		tmp = x * -y;
	} else if (x <= -1.9e-39) {
		tmp = x * z;
	} else if (x <= 1.7e-158) {
		tmp = y;
	} else if (x <= 5.7e-105) {
		tmp = x * z;
	} else if (x <= 1.6e-15) {
		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 <= (-1.82d+208)) then
        tmp = x * z
    else if (x <= (-1.6d+113)) then
        tmp = x * -y
    else if (x <= (-1.9d-39)) then
        tmp = x * z
    else if (x <= 1.7d-158) then
        tmp = y
    else if (x <= 5.7d-105) then
        tmp = x * z
    else if (x <= 1.6d-15) 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 <= -1.82e+208) {
		tmp = x * z;
	} else if (x <= -1.6e+113) {
		tmp = x * -y;
	} else if (x <= -1.9e-39) {
		tmp = x * z;
	} else if (x <= 1.7e-158) {
		tmp = y;
	} else if (x <= 5.7e-105) {
		tmp = x * z;
	} else if (x <= 1.6e-15) {
		tmp = y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.82e+208:
		tmp = x * z
	elif x <= -1.6e+113:
		tmp = x * -y
	elif x <= -1.9e-39:
		tmp = x * z
	elif x <= 1.7e-158:
		tmp = y
	elif x <= 5.7e-105:
		tmp = x * z
	elif x <= 1.6e-15:
		tmp = y
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.82e+208)
		tmp = Float64(x * z);
	elseif (x <= -1.6e+113)
		tmp = Float64(x * Float64(-y));
	elseif (x <= -1.9e-39)
		tmp = Float64(x * z);
	elseif (x <= 1.7e-158)
		tmp = y;
	elseif (x <= 5.7e-105)
		tmp = Float64(x * z);
	elseif (x <= 1.6e-15)
		tmp = y;
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.82e+208)
		tmp = x * z;
	elseif (x <= -1.6e+113)
		tmp = x * -y;
	elseif (x <= -1.9e-39)
		tmp = x * z;
	elseif (x <= 1.7e-158)
		tmp = y;
	elseif (x <= 5.7e-105)
		tmp = x * z;
	elseif (x <= 1.6e-15)
		tmp = y;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.82e+208], N[(x * z), $MachinePrecision], If[LessEqual[x, -1.6e+113], N[(x * (-y)), $MachinePrecision], If[LessEqual[x, -1.9e-39], N[(x * z), $MachinePrecision], If[LessEqual[x, 1.7e-158], y, If[LessEqual[x, 5.7e-105], N[(x * z), $MachinePrecision], If[LessEqual[x, 1.6e-15], y, N[(x * z), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.81999999999999999e208 or -1.5999999999999999e113 < x < -1.9000000000000001e-39 or 1.7e-158 < x < 5.69999999999999963e-105 or 1.6e-15 < x

   1. Initial program 95.6%

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

   2. Taylor expanded in y around 0 61.1%

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

   if -1.81999999999999999e208 < x < -1.5999999999999999e113

   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. sub-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 73.7%

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

      1. mul-1-neg 73.7%

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

      2. distribute-rgt-neg-out 73.7%

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

   7. Simplified 73.7%

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

   if -1.9000000000000001e-39 < x < 1.7e-158 or 5.69999999999999963e-105 < x < 1.6e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 80.0%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.82 \cdot 10^{+208}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{+113}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-39}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-158}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-105}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-15}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 3: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(z - y\right)\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-158}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-105}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-15}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z y))))
   (if (<= x -9.8e-39)
     t_0
     (if (<= x 1.7e-158)
       y
       (if (<= x 5.2e-105) (* x z) (if (<= x 1.65e-15) y t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (z - y);
	double tmp;
	if (x <= -9.8e-39) {
		tmp = t_0;
	} else if (x <= 1.7e-158) {
		tmp = y;
	} else if (x <= 5.2e-105) {
		tmp = x * z;
	} else if (x <= 1.65e-15) {
		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 <= (-9.8d-39)) then
        tmp = t_0
    else if (x <= 1.7d-158) then
        tmp = y
    else if (x <= 5.2d-105) then
        tmp = x * z
    else if (x <= 1.65d-15) 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 <= -9.8e-39) {
		tmp = t_0;
	} else if (x <= 1.7e-158) {
		tmp = y;
	} else if (x <= 5.2e-105) {
		tmp = x * z;
	} else if (x <= 1.65e-15) {
		tmp = y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (z - y)
	tmp = 0
	if x <= -9.8e-39:
		tmp = t_0
	elif x <= 1.7e-158:
		tmp = y
	elif x <= 5.2e-105:
		tmp = x * z
	elif x <= 1.65e-15:
		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 <= -9.8e-39)
		tmp = t_0;
	elseif (x <= 1.7e-158)
		tmp = y;
	elseif (x <= 5.2e-105)
		tmp = Float64(x * z);
	elseif (x <= 1.65e-15)
		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 <= -9.8e-39)
		tmp = t_0;
	elseif (x <= 1.7e-158)
		tmp = y;
	elseif (x <= 5.2e-105)
		tmp = x * z;
	elseif (x <= 1.65e-15)
		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, -9.8e-39], t$95$0, If[LessEqual[x, 1.7e-158], y, If[LessEqual[x, 5.2e-105], N[(x * z), $MachinePrecision], If[LessEqual[x, 1.65e-15], y, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.79999999999999947e-39 or 1.65e-15 < x

   1. Initial program 95.9%

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

   2. Taylor expanded in x around inf 96.0%

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

      1. mul-1-neg 96.0%

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

      2. sub-neg 96.0%

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

   4. Simplified 96.0%

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

   if -9.79999999999999947e-39 < x < 1.7e-158 or 5.1999999999999997e-105 < x < 1.65e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 80.0%

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

   if 1.7e-158 < x < 5.1999999999999997e-105

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 74.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-39}:\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-158}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-105}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-15}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z - y\right)\\ \end{array} \]

Alternative 4: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(z - y\right)\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-158}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-105}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 7400:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z y))))
   (if (<= x -1.32e-39)
     t_0
     (if (<= x 1.7e-158)
       y
       (if (<= x 4.7e-105) (* x z) (if (<= x 7400.0) (* y (- 1.0 x)) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (z - y);
	double tmp;
	if (x <= -1.32e-39) {
		tmp = t_0;
	} else if (x <= 1.7e-158) {
		tmp = y;
	} else if (x <= 4.7e-105) {
		tmp = x * z;
	} else if (x <= 7400.0) {
		tmp = y * (1.0 - x);
	} 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 <= (-1.32d-39)) then
        tmp = t_0
    else if (x <= 1.7d-158) then
        tmp = y
    else if (x <= 4.7d-105) then
        tmp = x * z
    else if (x <= 7400.0d0) then
        tmp = y * (1.0d0 - x)
    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 <= -1.32e-39) {
		tmp = t_0;
	} else if (x <= 1.7e-158) {
		tmp = y;
	} else if (x <= 4.7e-105) {
		tmp = x * z;
	} else if (x <= 7400.0) {
		tmp = y * (1.0 - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (z - y)
	tmp = 0
	if x <= -1.32e-39:
		tmp = t_0
	elif x <= 1.7e-158:
		tmp = y
	elif x <= 4.7e-105:
		tmp = x * z
	elif x <= 7400.0:
		tmp = y * (1.0 - x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(z - y))
	tmp = 0.0
	if (x <= -1.32e-39)
		tmp = t_0;
	elseif (x <= 1.7e-158)
		tmp = y;
	elseif (x <= 4.7e-105)
		tmp = Float64(x * z);
	elseif (x <= 7400.0)
		tmp = Float64(y * Float64(1.0 - x));
	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 <= -1.32e-39)
		tmp = t_0;
	elseif (x <= 1.7e-158)
		tmp = y;
	elseif (x <= 4.7e-105)
		tmp = x * z;
	elseif (x <= 7400.0)
		tmp = y * (1.0 - x);
	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, -1.32e-39], t$95$0, If[LessEqual[x, 1.7e-158], y, If[LessEqual[x, 4.7e-105], N[(x * z), $MachinePrecision], If[LessEqual[x, 7400.0], N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.31999999999999997e-39 or 7400 < x

   1. Initial program 95.8%

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

   2. Taylor expanded in x around inf 97.1%

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

      1. mul-1-neg 97.1%

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

      2. sub-neg 97.1%

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

   4. Simplified 97.1%

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

   if -1.31999999999999997e-39 < x < 1.7e-158

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 80.2%

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

   if 1.7e-158 < x < 4.69999999999999986e-105

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 74.3%

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

   if 4.69999999999999986e-105 < x < 7400

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 78.0%

      \[\leadsto \color{blue}{y \cdot \left(1 - x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{-39}:\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-158}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-105}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 7400:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z - y\right)\\ \end{array} \]

Alternative 5: 60.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-39}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-158}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-105}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-15}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.7e-39)
   (* x z)
   (if (<= x 1.7e-158)
     y
     (if (<= x 4.7e-105) (* x z) (if (<= x 1.55e-15) y (* x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.7e-39) {
		tmp = x * z;
	} else if (x <= 1.7e-158) {
		tmp = y;
	} else if (x <= 4.7e-105) {
		tmp = x * z;
	} else if (x <= 1.55e-15) {
		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 <= (-4.7d-39)) then
        tmp = x * z
    else if (x <= 1.7d-158) then
        tmp = y
    else if (x <= 4.7d-105) then
        tmp = x * z
    else if (x <= 1.55d-15) 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 <= -4.7e-39) {
		tmp = x * z;
	} else if (x <= 1.7e-158) {
		tmp = y;
	} else if (x <= 4.7e-105) {
		tmp = x * z;
	} else if (x <= 1.55e-15) {
		tmp = y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.7e-39:
		tmp = x * z
	elif x <= 1.7e-158:
		tmp = y
	elif x <= 4.7e-105:
		tmp = x * z
	elif x <= 1.55e-15:
		tmp = y
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.7e-39)
		tmp = Float64(x * z);
	elseif (x <= 1.7e-158)
		tmp = y;
	elseif (x <= 4.7e-105)
		tmp = Float64(x * z);
	elseif (x <= 1.55e-15)
		tmp = y;
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.7e-39)
		tmp = x * z;
	elseif (x <= 1.7e-158)
		tmp = y;
	elseif (x <= 4.7e-105)
		tmp = x * z;
	elseif (x <= 1.55e-15)
		tmp = y;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.7e-39], N[(x * z), $MachinePrecision], If[LessEqual[x, 1.7e-158], y, If[LessEqual[x, 4.7e-105], N[(x * z), $MachinePrecision], If[LessEqual[x, 1.55e-15], y, N[(x * z), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.7000000000000002e-39 or 1.7e-158 < x < 4.69999999999999986e-105 or 1.5499999999999999e-15 < x

   1. Initial program 96.2%

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

   2. Taylor expanded in y around 0 56.8%

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

   if -4.7000000000000002e-39 < x < 1.7e-158 or 4.69999999999999986e-105 < x < 1.5499999999999999e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 80.0%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-39}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-158}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-105}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-15}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 6: 98.8% 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 95.7%

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

   2. Taylor expanded in x around inf 98.4%

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

      1. mul-1-neg 98.4%

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

      2. sub-neg 98.4%

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

   4. Simplified 98.4%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      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 98.5%

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

4. Final simplification 98.4%

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

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 97.6%

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

   1. *-commutative 97.6%

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

   2. distribute-lft-out-- 97.6%

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

   3. *-rgt-identity 97.6%

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

   4. cancel-sign-sub-inv 97.6%

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

   5. +-commutative 97.6%

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

   6. +-commutative 97.6%

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

   7. associate-+l+ 97.6%

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

   8. +-commutative 97.6%

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

   9. *-commutative 97.6%

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

   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: 36.1% 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 97.6%

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

2. Taylor expanded in x around 0 34.3%

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

3. Final simplification 34.3%

   \[\leadsto y \]

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 2023293 
(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)))