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

Percentage Accurate: 97.8% → 100.0%

Time: 8.3s

Alternatives: 5

Speedup: 1.7×

• 20.1% 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, 1.7× 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 96.8%

   \[\left(1 - x\right) \cdot y + x \cdot z \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. mul-1-neg N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. mul-1-neg N/A

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

   6. associate-+r+ N/A

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

   7. mul-1-neg N/A

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

   8. unsub-neg N/A

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

   9. associate-+l- N/A

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

   10. distribute-lft-out-- N/A

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

   11. unsub-neg N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. mul-1-neg N/A

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

   15. +-commutative N/A

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

   16. mul-1-neg N/A

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

   17. distribute-rgt-neg-in N/A

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

   18. lower-fma.f64 N/A

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

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, z - y, y\right)} \]
6. Add Preprocessing

Alternative 2: 61.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+201}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{+41}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-37}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.9e+201)
   (* x z)
   (if (<= x -1.6e+41)
     (- (* x y))
     (if (<= x -4e-37) (* x z) (if (<= x 1.5e-66) (fma y x y) (* x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.9e+201) {
		tmp = x * z;
	} else if (x <= -1.6e+41) {
		tmp = -(x * y);
	} else if (x <= -4e-37) {
		tmp = x * z;
	} else if (x <= 1.5e-66) {
		tmp = fma(y, x, y);
	} else {
		tmp = x * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.9e+201)
		tmp = Float64(x * z);
	elseif (x <= -1.6e+41)
		tmp = Float64(-Float64(x * y));
	elseif (x <= -4e-37)
		tmp = Float64(x * z);
	elseif (x <= 1.5e-66)
		tmp = fma(y, x, y);
	else
		tmp = Float64(x * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.9e+201], N[(x * z), $MachinePrecision], If[LessEqual[x, -1.6e+41], (-N[(x * y), $MachinePrecision]), If[LessEqual[x, -4e-37], N[(x * z), $MachinePrecision], If[LessEqual[x, 1.5e-66], N[(y * x + y), $MachinePrecision], N[(x * z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.89999999999999998e201 or -1.60000000000000005e41 < x < -4.00000000000000027e-37 or 1.5000000000000001e-66 < x

   1. Initial program 94.7%

      \[\left(1 - x\right) \cdot y + x \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot z} \]
   4. Step-by-step derivation

      1. lower-*.f64 61.9

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

   5. Applied rewrites 61.9%

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

   if -1.89999999999999998e201 < x < -1.60000000000000005e41

   1. Initial program 95.3%

      \[\left(1 - x\right) \cdot y + x \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 62.7

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

   8. Applied rewrites 62.7%

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

   if -4.00000000000000027e-37 < x < 1.5000000000000001e-66

   1. Initial program 100.0%

      \[\left(1 - x\right) \cdot y + x \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{1 \cdot y - x \cdot y} \]

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

      \[\leadsto \color{blue}{y - y \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft1-in N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-lft-neg-out N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 75.1

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

   7. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, x, y\right)} \]
   8. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 75.1

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

   9. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+201}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{+41}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-37}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(z - y\right)\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{-37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z y))))
   (if (<= x -3.6e-37) t_0 (if (<= x 1.5e-66) (fma y x y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (z - y);
	double tmp;
	if (x <= -3.6e-37) {
		tmp = t_0;
	} else if (x <= 1.5e-66) {
		tmp = fma(y, x, 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 <= -3.6e-37)
		tmp = t_0;
	elseif (x <= 1.5e-66)
		tmp = fma(y, x, y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.6e-37], t$95$0, If[LessEqual[x, 1.5e-66], N[(y * x + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000007e-37 or 1.5000000000000001e-66 < x

   1. Initial program 94.9%

      \[\left(1 - x\right) \cdot y + x \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower--.f64 95.9

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

   5. Applied rewrites 95.9%

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

   if -3.60000000000000007e-37 < x < 1.5000000000000001e-66

   1. Initial program 100.0%

      \[\left(1 - x\right) \cdot y + x \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{1 \cdot y - x \cdot y} \]

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

      \[\leadsto \color{blue}{y - y \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft1-in N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-lft-neg-out N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 75.1

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

   7. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, x, y\right)} \]
   8. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 75.1

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

   9. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 60.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-37}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4e-37) (* x z) (if (<= x 1.5e-66) (fma y x y) (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4e-37) {
		tmp = x * z;
	} else if (x <= 1.5e-66) {
		tmp = fma(y, x, y);
	} else {
		tmp = x * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4e-37)
		tmp = Float64(x * z);
	elseif (x <= 1.5e-66)
		tmp = fma(y, x, y);
	else
		tmp = Float64(x * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4e-37], N[(x * z), $MachinePrecision], If[LessEqual[x, 1.5e-66], N[(y * x + y), $MachinePrecision], N[(x * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000027e-37 or 1.5000000000000001e-66 < x

   1. Initial program 94.9%

      \[\left(1 - x\right) \cdot y + x \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot z} \]
   4. Step-by-step derivation

      1. lower-*.f64 56.4

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

   5. Applied rewrites 56.4%

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

   if -4.00000000000000027e-37 < x < 1.5000000000000001e-66

   1. Initial program 100.0%

      \[\left(1 - x\right) \cdot y + x \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{1 \cdot y - x \cdot y} \]

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

      \[\leadsto \color{blue}{y - y \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft1-in N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-lft-neg-out N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 75.1

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

   7. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, x, y\right)} \]
   8. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 75.1

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

   9. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 40.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ x \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x z))

C

double code(double x, double y, double z) {
	return 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 = x * z
end function

Java

public static double code(double x, double y, double z) {
	return x * z;
}

Python

def code(x, y, z):
	return x * z

Julia

function code(x, y, z)
	return Float64(x * z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * z;
end

Wolfram

code[x_, y_, z_] := N[(x * z), $MachinePrecision]

Derivation

1. Initial program 96.8%

   \[\left(1 - x\right) \cdot y + x \cdot z \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x \cdot z} \]
4. Step-by-step derivation

   1. lower-*.f64 44.7

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

5. Applied rewrites 44.7%

   \[\leadsto \color{blue}{x \cdot z} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.4× 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 2024216 
(FPCore (x y z)
  :name "Diagrams.Color.HSV:lerp  from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (! :herbie-platform default (- y (* x (- y z))))

  (+ (* (- 1.0 x) y) (* x z)))