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

Percentage Accurate: 97.9% → 100.0%

Time: 5.5s

Alternatives: 7

Speedup: 1.7×

• 20.9% 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, 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 97.3%

   \[x \cdot y + \left(1 - x\right) \cdot z \]
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. *-commutative N/A

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

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

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

   5. *-lft-identity N/A

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

   6. associate-+l- N/A

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

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

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

   8. unsub-neg N/A

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

   9. mul-1-neg N/A

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

   10. sub-neg N/A

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

   11. mul-1-neg N/A

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

   12. +-commutative N/A

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

   13. mul-1-neg N/A

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

   14. *-commutative N/A

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

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

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

   16. lower-fma.f64 N/A

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

   17. +-commutative N/A

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

   18. distribute-neg-in N/A

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

   19. mul-1-neg N/A

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

   20. remove-double-neg N/A

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

   21. unsub-neg N/A

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

   22. lower--.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 80.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+52}:\\ \;\;\;\;\left(1 - x\right) \cdot z\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-36}:\\ \;\;\;\;\left(y - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8.5e+52)
   (* (- 1.0 x) z)
   (if (<= z 7e-36) (* (- y z) x) (fma (- z) x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.5e+52) {
		tmp = (1.0 - x) * z;
	} else if (z <= 7e-36) {
		tmp = (y - z) * x;
	} else {
		tmp = fma(-z, x, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8.5e+52)
		tmp = Float64(Float64(1.0 - x) * z);
	elseif (z <= 7e-36)
		tmp = Float64(Float64(y - z) * x);
	else
		tmp = fma(Float64(-z), x, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8.5e+52], N[(N[(1.0 - x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 7e-36], N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision], N[((-z) * x + z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.49999999999999994e52

   1. Initial program 94.8%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   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 87.8

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

   5. Applied rewrites 87.8%

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

   if -8.49999999999999994e52 < z < 6.9999999999999999e-36

   1. Initial program 99.1%

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

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      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 83.2

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

   5. Applied rewrites 83.2%

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

   if 6.9999999999999999e-36 < z

   1. Initial program 96.3%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   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 89.4

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

   5. Applied rewrites 89.4%

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

   7. Applied rewrites 89.4%

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

Alternative 3: 80.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - x\right) \cdot z\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-36}:\\ \;\;\;\;\left(y - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- 1.0 x) z)))
   (if (<= z -8.5e+52) t_0 (if (<= z 7e-36) (* (- y z) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (1.0 - x) * z;
	double tmp;
	if (z <= -8.5e+52) {
		tmp = t_0;
	} else if (z <= 7e-36) {
		tmp = (y - z) * 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 = (1.0d0 - x) * z
    if (z <= (-8.5d+52)) then
        tmp = t_0
    else if (z <= 7d-36) then
        tmp = (y - z) * 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 = (1.0 - x) * z;
	double tmp;
	if (z <= -8.5e+52) {
		tmp = t_0;
	} else if (z <= 7e-36) {
		tmp = (y - z) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (1.0 - x) * z
	tmp = 0
	if z <= -8.5e+52:
		tmp = t_0
	elif z <= 7e-36:
		tmp = (y - z) * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(1.0 - x) * z)
	tmp = 0.0
	if (z <= -8.5e+52)
		tmp = t_0;
	elseif (z <= 7e-36)
		tmp = Float64(Float64(y - z) * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (1.0 - x) * z;
	tmp = 0.0;
	if (z <= -8.5e+52)
		tmp = t_0;
	elseif (z <= 7e-36)
		tmp = (y - z) * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -8.5e+52], t$95$0, If[LessEqual[z, 7e-36], N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.49999999999999994e52 or 6.9999999999999999e-36 < z

   1. Initial program 95.7%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   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 88.8

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

   5. Applied rewrites 88.8%

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

   if -8.49999999999999994e52 < z < 6.9999999999999999e-36

   1. Initial program 99.1%

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

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      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 83.2

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

   5. Applied rewrites 83.2%

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

Alternative 4: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - x\right) \cdot z\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{-51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-36}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- 1.0 x) z)))
   (if (<= z -3.8e-51) t_0 (if (<= z 2.3e-36) (* y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (1.0 - x) * z;
	double tmp;
	if (z <= -3.8e-51) {
		tmp = t_0;
	} else if (z <= 2.3e-36) {
		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 = (1.0d0 - x) * z
    if (z <= (-3.8d-51)) then
        tmp = t_0
    else if (z <= 2.3d-36) 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 = (1.0 - x) * z;
	double tmp;
	if (z <= -3.8e-51) {
		tmp = t_0;
	} else if (z <= 2.3e-36) {
		tmp = y * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (1.0 - x) * z
	tmp = 0
	if z <= -3.8e-51:
		tmp = t_0
	elif z <= 2.3e-36:
		tmp = y * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(1.0 - x) * z)
	tmp = 0.0
	if (z <= -3.8e-51)
		tmp = t_0;
	elseif (z <= 2.3e-36)
		tmp = Float64(y * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (1.0 - x) * z;
	tmp = 0.0;
	if (z <= -3.8e-51)
		tmp = t_0;
	elseif (z <= 2.3e-36)
		tmp = y * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3.8e-51], t$95$0, If[LessEqual[z, 2.3e-36], N[(y * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.80000000000000003e-51 or 2.29999999999999996e-36 < z

   1. Initial program 95.7%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   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 83.4

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

   5. Applied rewrites 83.4%

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

   if -3.80000000000000003e-51 < z < 2.29999999999999996e-36

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   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 75.6

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

   5. Applied rewrites 75.6%

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

Alternative 5: 61.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-72}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7.2e-72) (* y x) (if (<= x 1.0) (* 1.0 z) (* (- z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.2e-72) {
		tmp = y * x;
	} else if (x <= 1.0) {
		tmp = 1.0 * z;
	} else {
		tmp = -z * 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 <= (-7.2d-72)) then
        tmp = y * x
    else if (x <= 1.0d0) then
        tmp = 1.0d0 * z
    else
        tmp = -z * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.2e-72) {
		tmp = y * x;
	} else if (x <= 1.0) {
		tmp = 1.0 * z;
	} else {
		tmp = -z * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7.2e-72:
		tmp = y * x
	elif x <= 1.0:
		tmp = 1.0 * z
	else:
		tmp = -z * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7.2e-72)
		tmp = Float64(y * x);
	elseif (x <= 1.0)
		tmp = Float64(1.0 * z);
	else
		tmp = Float64(Float64(-z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7.2e-72)
		tmp = y * x;
	elseif (x <= 1.0)
		tmp = 1.0 * z;
	else
		tmp = -z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.2e-72], N[(y * x), $MachinePrecision], If[LessEqual[x, 1.0], N[(1.0 * z), $MachinePrecision], N[((-z) * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.2e-72

   1. Initial program 98.7%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   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 58.5

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

   5. Applied rewrites 58.5%

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

   if -7.2e-72 < x < 1

   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 72.4

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.8%

      \[\leadsto 1 \cdot z \]

   if 1 < x

   1. Initial program 90.3%

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

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      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 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 62.9%

      \[\leadsto \left(-z\right) \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 60.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-72}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-61}:\\ \;\;\;\;1 \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7.2e-72) (* y x) (if (<= x 7.6e-61) (* 1.0 z) (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.2e-72) {
		tmp = y * x;
	} else if (x <= 7.6e-61) {
		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 <= (-7.2d-72)) then
        tmp = y * x
    else if (x <= 7.6d-61) 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 <= -7.2e-72) {
		tmp = y * x;
	} else if (x <= 7.6e-61) {
		tmp = 1.0 * z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7.2e-72:
		tmp = y * x
	elif x <= 7.6e-61:
		tmp = 1.0 * z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7.2e-72)
		tmp = Float64(y * x);
	elseif (x <= 7.6e-61)
		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 <= -7.2e-72)
		tmp = y * x;
	elseif (x <= 7.6e-61)
		tmp = 1.0 * z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.2e-72], N[(y * x), $MachinePrecision], If[LessEqual[x, 7.6e-61], N[(1.0 * z), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.2e-72 or 7.59999999999999961e-61 < x

   1. Initial program 95.5%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   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 49.8

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

   5. Applied rewrites 49.8%

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

   if -7.2e-72 < x < 7.59999999999999961e-61

   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 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.3%

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

Alternative 7: 41.2% 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 97.3%

   \[x \cdot y + \left(1 - x\right) \cdot z \]
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 40.3

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

5. Applied rewrites 40.3%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))