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

Percentage Accurate: 97.9% → 100.0%

Time: 8.1s

Alternatives: 8

Speedup: 1.3×

• 20.7% 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: 97.9% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z - y, 0 - x, z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (- z y) (- 0.0 x) z))

C

double code(double x, double y, double z) {
	return fma((z - y), (0.0 - x), z);
}

Julia

function code(x, y, z)
	return fma(Float64(z - y), Float64(0.0 - x), z)
end

Wolfram

code[x_, y_, z_] := N[(N[(z - y), $MachinePrecision] * N[(0.0 - x), $MachinePrecision] + z), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

   4. *-lft-identity N/A

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

   5. associate-+l- N/A

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

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(z, \color{blue}{\left(x \cdot z - x \cdot y\right)}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(z, \left(x \cdot \color{blue}{\left(z - y\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(x, \color{blue}{\left(z - y\right)}\right)\right) \]

   9. --lowering--.f64 100.0%

      \[\leadsto \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(z, \color{blue}{y}\right)\right)\right) \]

3. Simplified 100.0%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

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

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

   5. fma-define N/A

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

   6. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(z - y\right), \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}, z\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(z, y\right), \left(\mathsf{neg}\left(\color{blue}{x}\right)\right), z\right) \]

   8. neg-sub0 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(z, y\right), \left(0 - \color{blue}{x}\right), z\right) \]

   9. --lowering--.f64 100.0%

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(z, y\right), \mathsf{\_.f64}\left(0, \color{blue}{x}\right), z\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z - y, 0 - x, z\right)} \]
7. Add Preprocessing

Alternative 2: 60.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(0 - x\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+227}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- 0.0 x))))
   (if (<= x -9e-7) t_0 (if (<= x 1.0) z (if (<= x 1.1e+227) t_0 (* y x))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (0.0 - x);
	double tmp;
	if (x <= -9e-7) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = z;
	} else if (x <= 1.1e+227) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = z * (0.0d0 - x)
    if (x <= (-9d-7)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = z
    else if (x <= 1.1d+227) then
        tmp = t_0
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (0.0 - x);
	double tmp;
	if (x <= -9e-7) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = z;
	} else if (x <= 1.1e+227) {
		tmp = t_0;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (0.0 - x)
	tmp = 0
	if x <= -9e-7:
		tmp = t_0
	elif x <= 1.0:
		tmp = z
	elif x <= 1.1e+227:
		tmp = t_0
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(0.0 - x))
	tmp = 0.0
	if (x <= -9e-7)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = z;
	elseif (x <= 1.1e+227)
		tmp = t_0;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (0.0 - x);
	tmp = 0.0;
	if (x <= -9e-7)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = z;
	elseif (x <= 1.1e+227)
		tmp = t_0;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(0.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e-7], t$95$0, If[LessEqual[x, 1.0], z, If[LessEqual[x, 1.1e+227], t$95$0, N[(y * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.99999999999999959e-7 or 1 < x < 1.1000000000000001e227

   1. Initial program 97.9%

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

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \left(z - y\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z - \color{blue}{-1 \cdot y}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot y\right)\right)}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z + y\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y + \color{blue}{-1 \cdot z}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y + \left(\mathsf{neg}\left(z\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y - \color{blue}{z}\right)\right) \]

      14. --lowering--.f64 99.3%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 99.3%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. distribute-neg-in N/A

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

      5. sub-neg N/A

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

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

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

      7. /-rgt-identity N/A

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

      8. associate-/r/ N/A

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

      9. distribute-neg-frac2 N/A

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

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{z - y}\right)\right)}\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{z - y}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{-1}{\color{blue}{z} - y}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{\left(z - y\right)}\right)\right) \]

      14. --lowering--.f64 99.2%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(z, \color{blue}{y}\right)\right)\right) \]

   7. Applied egg-rr 99.2%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{-1}{z}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 63.1%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{z}\right)\right) \]

   10. Simplified 63.1%

       \[\leadsto \frac{x}{\color{blue}{\frac{-1}{z}}} \]
   11. Step-by-step derivation

       1. frac-2neg N/A

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

       2. distribute-frac-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\mathsf{neg}\left(\frac{-1}{z}\right)}\right) \]

       3. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\mathsf{neg}\left(\frac{-1}{z}\right)}\right)\right) \]

       4. distribute-neg-frac N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\frac{\mathsf{neg}\left(-1\right)}{z}}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\frac{1}{z}}\right)\right) \]

       6. associate-/r/ N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{1} \cdot z\right)\right) \]

       7. /-rgt-identity N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(x \cdot z\right)\right) \]

       8. *-lowering-*.f64 63.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, z\right)\right) \]

   12. Applied egg-rr 63.3%

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

   if -8.99999999999999959e-7 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 71.6%

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

   if 1.1000000000000001e227 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 85.2%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 85.2%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-7}:\\ \;\;\;\;z \cdot \left(0 - x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+227}:\\ \;\;\;\;z \cdot \left(0 - x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- y z))))
   (if (<= x -1.0) t_0 (if (<= x 1.0) (+ z (* y x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y - z);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = z + (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 = x * (y - z)
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = z + (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 = x * (y - z);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = z + (y * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 98.1%

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

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \left(z - y\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z - \color{blue}{-1 \cdot y}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot y\right)\right)}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z + y\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y + \color{blue}{-1 \cdot z}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y + \left(\mathsf{neg}\left(z\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y - \color{blue}{z}\right)\right) \]

      14. --lowering--.f64 99.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 99.4%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{z}\right) \]
   4. Step-by-step derivation

   5. Simplified 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;z + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y - z\right)\\ \mathbf{if}\;x \leq -3.3 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-50}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- y z))))
   (if (<= x -3.3e-12) t_0 (if (<= x 3.2e-50) (* z (- 1.0 x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y - z);
	double tmp;
	if (x <= -3.3e-12) {
		tmp = t_0;
	} else if (x <= 3.2e-50) {
		tmp = z * (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 * (y - z)
    if (x <= (-3.3d-12)) then
        tmp = t_0
    else if (x <= 3.2d-50) then
        tmp = z * (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 * (y - z);
	double tmp;
	if (x <= -3.3e-12) {
		tmp = t_0;
	} else if (x <= 3.2e-50) {
		tmp = z * (1.0 - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y - z)
	tmp = 0
	if x <= -3.3e-12:
		tmp = t_0
	elif x <= 3.2e-50:
		tmp = z * (1.0 - x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y - z))
	tmp = 0.0
	if (x <= -3.3e-12)
		tmp = t_0;
	elseif (x <= 3.2e-50)
		tmp = Float64(z * Float64(1.0 - x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y - z);
	tmp = 0.0;
	if (x <= -3.3e-12)
		tmp = t_0;
	elseif (x <= 3.2e-50)
		tmp = z * (1.0 - x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.3e-12], t$95$0, If[LessEqual[x, 3.2e-50], N[(z * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.3000000000000001e-12 or 3.2e-50 < x

   1. Initial program 98.3%

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

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \left(z - y\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z - \color{blue}{-1 \cdot y}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot y\right)\right)}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z + y\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y + \color{blue}{-1 \cdot z}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y + \left(\mathsf{neg}\left(z\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y - \color{blue}{z}\right)\right) \]

      14. --lowering--.f64 94.6%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 94.6%

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

   if -3.3000000000000001e-12 < x < 3.2e-50

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(1 - x\right)}\right) \]

      2. --lowering--.f64 75.3%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(1, \color{blue}{x}\right)\right) \]

   5. Simplified 75.3%

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

Alternative 5: 84.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y - z\right)\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-53}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- y z))))
   (if (<= x -1.32e-11) t_0 (if (<= x 4e-53) z t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y - z);
	double tmp;
	if (x <= -1.32e-11) {
		tmp = t_0;
	} else if (x <= 4e-53) {
		tmp = 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 = x * (y - z)
    if (x <= (-1.32d-11)) then
        tmp = t_0
    else if (x <= 4d-53) then
        tmp = 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 = x * (y - z);
	double tmp;
	if (x <= -1.32e-11) {
		tmp = t_0;
	} else if (x <= 4e-53) {
		tmp = z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y - z)
	tmp = 0
	if x <= -1.32e-11:
		tmp = t_0
	elif x <= 4e-53:
		tmp = z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y - z))
	tmp = 0.0
	if (x <= -1.32e-11)
		tmp = t_0;
	elseif (x <= 4e-53)
		tmp = z;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y - z);
	tmp = 0.0;
	if (x <= -1.32e-11)
		tmp = t_0;
	elseif (x <= 4e-53)
		tmp = z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.32e-11], t$95$0, If[LessEqual[x, 4e-53], z, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.32e-11 or 4.00000000000000012e-53 < x

   1. Initial program 98.3%

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

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \left(z - y\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z - \color{blue}{-1 \cdot y}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot y\right)\right)}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z + y\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y + \color{blue}{-1 \cdot z}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y + \left(\mathsf{neg}\left(z\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y - \color{blue}{z}\right)\right) \]

      14. --lowering--.f64 94.6%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 94.6%

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

   if -1.32e-11 < x < 4.00000000000000012e-53

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 75.2%

      \[\leadsto \color{blue}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 59.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.55 \cdot 10^{-12}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-51}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.55e-12) (* y x) (if (<= x 8.5e-51) z (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.55e-12) {
		tmp = y * x;
	} else if (x <= 8.5e-51) {
		tmp = 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 <= (-3.55d-12)) then
        tmp = y * x
    else if (x <= 8.5d-51) then
        tmp = 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 <= -3.55e-12) {
		tmp = y * x;
	} else if (x <= 8.5e-51) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.55e-12:
		tmp = y * x
	elif x <= 8.5e-51:
		tmp = z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.55e-12)
		tmp = Float64(y * x);
	elseif (x <= 8.5e-51)
		tmp = z;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.55e-12)
		tmp = y * x;
	elseif (x <= 8.5e-51)
		tmp = z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.55e-12], N[(y * x), $MachinePrecision], If[LessEqual[x, 8.5e-51], z, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.55e-12 or 8.50000000000000036e-51 < x

   1. Initial program 98.3%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 47.2%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 47.2%

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

   if -3.55e-12 < x < 8.50000000000000036e-51

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 75.2%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.55 \cdot 10^{-12}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-51}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

   4. *-lft-identity N/A

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

   5. associate-+l- N/A

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

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(z, \color{blue}{\left(x \cdot z - x \cdot y\right)}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(z, \left(x \cdot \color{blue}{\left(z - y\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(x, \color{blue}{\left(z - y\right)}\right)\right) \]

   9. --lowering--.f64 100.0%

      \[\leadsto \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(z, \color{blue}{y}\right)\right)\right) \]

3. Simplified 100.0%

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

5. Final simplification 100.0%

   \[\leadsto z - \left(z - y\right) \cdot x \]
6. Add Preprocessing

Alternative 8: 35.8% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 42.2%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))