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

Percentage Accurate: 97.8% → 100.0%

Time: 7.8s

Alternatives: 5

Speedup: 1.7×

• 21.1% 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: 97.8% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-in N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. mul-1-neg N/A

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

   11. remove-double-neg N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 100.0

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

5. Simplified 100.0%

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

Alternative 2: 61.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-14}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(z, x, z\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+21}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8e-14)
   (* x y)
   (if (<= x 4.2e-19) (fma z x z) (if (<= x 3e+21) (* x y) (* x (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8e-14) {
		tmp = x * y;
	} else if (x <= 4.2e-19) {
		tmp = fma(z, x, z);
	} else if (x <= 3e+21) {
		tmp = x * y;
	} else {
		tmp = x * -z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8e-14)
		tmp = Float64(x * y);
	elseif (x <= 4.2e-19)
		tmp = fma(z, x, z);
	elseif (x <= 3e+21)
		tmp = Float64(x * y);
	else
		tmp = Float64(x * Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8e-14], N[(x * y), $MachinePrecision], If[LessEqual[x, 4.2e-19], N[(z * x + z), $MachinePrecision], If[LessEqual[x, 3e+21], N[(x * y), $MachinePrecision], N[(x * (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.99999999999999999e-14 or 4.1999999999999998e-19 < x < 3e21

   1. Initial program 93.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 56.2

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

   5. Simplified 56.2%

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

   if -7.99999999999999999e-14 < x < 4.1999999999999998e-19

   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. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 79.0

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

   5. Simplified 79.0%

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

      1. *-commutative N/A

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

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

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

      3. lift-neg.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft1-in N/A

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

      6. lift-neg.f64 N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 79.0

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

   7. Applied egg-rr 79.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 79.0

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

   9. Applied egg-rr 79.0%

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

   if 3e21 < x

   1. Initial program 95.1%

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

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 75.2

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

   8. Simplified 75.2%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-14}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(z, x, z\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+21}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y - z\right)\\ \mathbf{if}\;x \leq -3 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(z, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- y z))))
   (if (<= x -3e-19) t_0 (if (<= x 1.1e-12) (fma z x z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y - z);
	double tmp;
	if (x <= -3e-19) {
		tmp = t_0;
	} else if (x <= 1.1e-12) {
		tmp = fma(z, x, 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 <= -3e-19)
		tmp = t_0;
	elseif (x <= 1.1e-12)
		tmp = fma(z, x, z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3e-19], t$95$0, If[LessEqual[x, 1.1e-12], N[(z * x + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.99999999999999993e-19 or 1.09999999999999996e-12 < x

   1. Initial program 94.2%

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

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 99.6

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

   5. Simplified 99.6%

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

   if -2.99999999999999993e-19 < x < 1.09999999999999996e-12

   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. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 78.6

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

   5. Simplified 78.6%

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

      1. *-commutative N/A

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

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

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

      3. lift-neg.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft1-in N/A

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

      6. lift-neg.f64 N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 78.6

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

   7. Applied egg-rr 78.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 78.6

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

   9. Applied egg-rr 78.6%

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

Alternative 4: 61.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-13}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 3.65 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(z, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.4e-13) (* x y) (if (<= x 3.65e-19) (fma z x z) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.4e-13) {
		tmp = x * y;
	} else if (x <= 3.65e-19) {
		tmp = fma(z, x, z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.4e-13)
		tmp = Float64(x * y);
	elseif (x <= 3.65e-19)
		tmp = fma(z, x, z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.4e-13], N[(x * y), $MachinePrecision], If[LessEqual[x, 3.65e-19], N[(z * x + z), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.40000000000000021e-13 or 3.6499999999999998e-19 < x

   1. Initial program 94.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 43.4

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

   5. Simplified 43.4%

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

   if -5.40000000000000021e-13 < x < 3.6499999999999998e-19

   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. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 79.0

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

   5. Simplified 79.0%

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

      1. *-commutative N/A

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

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

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

      3. lift-neg.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft1-in N/A

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

      6. lift-neg.f64 N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 79.0

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

   7. Applied egg-rr 79.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 79.0

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

   9. Applied egg-rr 79.0%

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

Alternative 5: 42.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 32.9

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

5. Simplified 32.9%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))