Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3

Percentage Accurate: 98.4% → 100.0%

Time: 6.2s

Alternatives: 7

Speedup: 1.7×

• 20.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot y + \left(1 - x\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x y) (* (- 1.0 x) z)))

C

double code(double x, double y, double z) {
	return (x * y) + ((1.0 - 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 * y) + ((1.0d0 - x) * z)
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) + ((1.0 - x) * z);
}

Python

def code(x, y, z):
	return (x * y) + ((1.0 - x) * z)

Julia

function code(x, y, z)
	return Float64(Float64(x * y) + Float64(Float64(1.0 - x) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) + ((1.0 - x) * z);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Initial Program: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot y + \left(1 - x\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x y) (* (- 1.0 x) z)))

C

double code(double x, double y, double z) {
	return (x * y) + ((1.0 - 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 * y) + ((1.0d0 - x) * z)
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) + ((1.0 - x) * z);
}

Python

def code(x, y, z):
	return (x * y) + ((1.0 - x) * z)

Julia

function code(x, y, z)
	return Float64(Float64(x * y) + Float64(Float64(1.0 - x) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) + ((1.0 - x) * z);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y - z, x, z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (- y z) x z))

C

double code(double x, double y, double z) {
	return fma((y - z), x, z);
}

Julia

function code(x, y, z)
	return fma(Float64(y - z), x, z)
end

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. sub-neg N/A

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

   4. mul-1-neg N/A

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

   5. +-commutative N/A

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

   6. distribute-lft1-in N/A

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

   7. associate-*r* N/A

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

   8. +-commutative N/A

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

   9. mul-1-neg N/A

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

   10. unsub-neg N/A

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

   11. associate-+l- N/A

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

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

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

   13. unsub-neg N/A

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

   14. mul-1-neg N/A

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

   15. sub-neg N/A

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

   16. mul-1-neg N/A

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

   17. +-commutative N/A

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

   18. mul-1-neg N/A

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

   19. *-commutative N/A

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

   20. distribute-lft-neg-in N/A

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

   21. lower-fma.f64 N/A

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

5. Applied rewrites 100.0%

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

Alternative 2: 61.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot x\\ \mathbf{if}\;x \leq -2.25 \cdot 10^{+259}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-7}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-51}:\\ \;\;\;\;1 \cdot z\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{+252}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) x)))
   (if (<= x -2.25e+259)
     t_0
     (if (<= x -6.2e-7)
       (* y x)
       (if (<= x 6.8e-51) (* 1.0 z) (if (<= x 1.26e+252) (* y x) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = -z * x;
	double tmp;
	if (x <= -2.25e+259) {
		tmp = t_0;
	} else if (x <= -6.2e-7) {
		tmp = y * x;
	} else if (x <= 6.8e-51) {
		tmp = 1.0 * z;
	} else if (x <= 1.26e+252) {
		tmp = y * 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 = -z * x
    if (x <= (-2.25d+259)) then
        tmp = t_0
    else if (x <= (-6.2d-7)) then
        tmp = y * x
    else if (x <= 6.8d-51) then
        tmp = 1.0d0 * z
    else if (x <= 1.26d+252) then
        tmp = y * 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 = -z * x;
	double tmp;
	if (x <= -2.25e+259) {
		tmp = t_0;
	} else if (x <= -6.2e-7) {
		tmp = y * x;
	} else if (x <= 6.8e-51) {
		tmp = 1.0 * z;
	} else if (x <= 1.26e+252) {
		tmp = y * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z * x
	tmp = 0
	if x <= -2.25e+259:
		tmp = t_0
	elif x <= -6.2e-7:
		tmp = y * x
	elif x <= 6.8e-51:
		tmp = 1.0 * z
	elif x <= 1.26e+252:
		tmp = y * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) * x)
	tmp = 0.0
	if (x <= -2.25e+259)
		tmp = t_0;
	elseif (x <= -6.2e-7)
		tmp = Float64(y * x);
	elseif (x <= 6.8e-51)
		tmp = Float64(1.0 * z);
	elseif (x <= 1.26e+252)
		tmp = Float64(y * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z * x;
	tmp = 0.0;
	if (x <= -2.25e+259)
		tmp = t_0;
	elseif (x <= -6.2e-7)
		tmp = y * x;
	elseif (x <= 6.8e-51)
		tmp = 1.0 * z;
	elseif (x <= 1.26e+252)
		tmp = y * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * x), $MachinePrecision]}, If[LessEqual[x, -2.25e+259], t$95$0, If[LessEqual[x, -6.2e-7], N[(y * x), $MachinePrecision], If[LessEqual[x, 6.8e-51], N[(1.0 * z), $MachinePrecision], If[LessEqual[x, 1.26e+252], N[(y * x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2499999999999998e259 or 1.2599999999999999e252 < x

   1. Initial program 58.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot z\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 80.1%

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

   if -2.2499999999999998e259 < x < -6.1999999999999999e-7 or 6.80000000000000005e-51 < x < 1.2599999999999999e252

   1. Initial program 96.7%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if -6.1999999999999999e-7 < x < 6.80000000000000005e-51

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 73.6

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 73.0%

      \[\leadsto 1 \cdot z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - z\right) \cdot x\\ \mathbf{if}\;x \leq -0.19:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- y z) x)))
   (if (<= x -0.19) t_0 (if (<= x 6.8e-51) (fma (- z) x z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y - z) * x;
	double tmp;
	if (x <= -0.19) {
		tmp = t_0;
	} else if (x <= 6.8e-51) {
		tmp = fma(-z, x, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - z) * x)
	tmp = 0.0
	if (x <= -0.19)
		tmp = t_0;
	elseif (x <= 6.8e-51)
		tmp = fma(Float64(-z), x, z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -0.19], t$95$0, If[LessEqual[x, 6.8e-51], N[((-z) * x + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.19 or 6.80000000000000005e-51 < x

   1. Initial program 91.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if -0.19 < x < 6.80000000000000005e-51

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 72.8

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

   5. Applied rewrites 72.8%

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

   7. Applied rewrites 72.8%

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

Alternative 4: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - z\right) \cdot x\\ \mathbf{if}\;x \leq -0.19:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-51}:\\ \;\;\;\;\left(1 - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- y z) x)))
   (if (<= x -0.19) t_0 (if (<= x 6.8e-51) (* (- 1.0 x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y - z) * x;
	double tmp;
	if (x <= -0.19) {
		tmp = t_0;
	} else if (x <= 6.8e-51) {
		tmp = (1.0 - x) * z;
	} 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 = (y - z) * x
    if (x <= (-0.19d0)) then
        tmp = t_0
    else if (x <= 6.8d-51) then
        tmp = (1.0d0 - x) * z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - z) * x;
	double tmp;
	if (x <= -0.19) {
		tmp = t_0;
	} else if (x <= 6.8e-51) {
		tmp = (1.0 - x) * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - z) * x
	tmp = 0
	if x <= -0.19:
		tmp = t_0
	elif x <= 6.8e-51:
		tmp = (1.0 - x) * z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - z) * x)
	tmp = 0.0
	if (x <= -0.19)
		tmp = t_0;
	elseif (x <= 6.8e-51)
		tmp = Float64(Float64(1.0 - x) * z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - z) * x;
	tmp = 0.0;
	if (x <= -0.19)
		tmp = t_0;
	elseif (x <= 6.8e-51)
		tmp = (1.0 - x) * z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -0.19], t$95$0, If[LessEqual[x, 6.8e-51], N[(N[(1.0 - x), $MachinePrecision] * z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.19 or 6.80000000000000005e-51 < x

   1. Initial program 91.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if -0.19 < x < 6.80000000000000005e-51

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 72.8

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

   5. Applied rewrites 72.8%

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

Alternative 5: 72.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+192}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-39}:\\ \;\;\;\;\left(1 - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.2e+192) (* y x) (if (<= y 4.4e-39) (* (- 1.0 x) z) (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e+192) {
		tmp = y * x;
	} else if (y <= 4.4e-39) {
		tmp = (1.0 - x) * z;
	} else {
		tmp = y * 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 (y <= (-2.2d+192)) then
        tmp = y * x
    else if (y <= 4.4d-39) then
        tmp = (1.0d0 - x) * z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e+192) {
		tmp = y * x;
	} else if (y <= 4.4e-39) {
		tmp = (1.0 - x) * z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.2e+192:
		tmp = y * x
	elif y <= 4.4e-39:
		tmp = (1.0 - x) * z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.2e+192)
		tmp = Float64(y * x);
	elseif (y <= 4.4e-39)
		tmp = Float64(Float64(1.0 - x) * z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.2e+192)
		tmp = y * x;
	elseif (y <= 4.4e-39)
		tmp = (1.0 - x) * z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.2e+192], N[(y * x), $MachinePrecision], If[LessEqual[y, 4.4e-39], N[(N[(1.0 - x), $MachinePrecision] * z), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2000000000000001e192 or 4.40000000000000002e-39 < y

   1. Initial program 95.2%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if -2.2000000000000001e192 < y < 4.40000000000000002e-39

   1. Initial program 96.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 77.5

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

   5. Applied rewrites 77.5%

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

Alternative 6: 61.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-7}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-51}:\\ \;\;\;\;1 \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.2e-7) (* y x) (if (<= x 6.8e-51) (* 1.0 z) (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.2e-7) {
		tmp = y * x;
	} else if (x <= 6.8e-51) {
		tmp = 1.0 * z;
	} else {
		tmp = y * 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 <= (-6.2d-7)) then
        tmp = y * x
    else if (x <= 6.8d-51) then
        tmp = 1.0d0 * z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.2e-7) {
		tmp = y * x;
	} else if (x <= 6.8e-51) {
		tmp = 1.0 * z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6.2e-7:
		tmp = y * x
	elif x <= 6.8e-51:
		tmp = 1.0 * z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6.2e-7)
		tmp = Float64(y * x);
	elseif (x <= 6.8e-51)
		tmp = Float64(1.0 * z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -6.2e-7)
		tmp = y * x;
	elseif (x <= 6.8e-51)
		tmp = 1.0 * z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.2e-7], N[(y * x), $MachinePrecision], If[LessEqual[x, 6.8e-51], N[(1.0 * z), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.1999999999999999e-7 or 6.80000000000000005e-51 < x

   1. Initial program 92.0%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

   if -6.1999999999999999e-7 < x < 6.80000000000000005e-51

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 73.6

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 73.0%

      \[\leadsto 1 \cdot z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 42.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 44.0

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

5. Applied rewrites 44.0%

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

Reproduce

herbie shell --seed 2024248 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))