Numeric.Histogram:binBounds from Chart-1.5.3

Percentage Accurate: 92.6% → 97.7%

Time: 9.7s

Alternatives: 10

Speedup: 1.1×

• 24.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x + (((y - x) * z) / t);
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x + (((y - x) * z) / t);
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ x + \frac{y - x}{\frac{t}{z}} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ x (/ (- y x) (/ t z))))

C

double code(double x, double y, double z, double t) {
	return x + ((y - x) / (t / z));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y - x) / (t / z))
end function

Java

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

Python

def code(x, y, z, t):
	return x + ((y - x) / (t / z))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y - x) / Float64(t / z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + ((y - x) / (t / z));
end

Wolfram

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

Derivation

1. Initial program 91.8%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. remove-double-neg N/A

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

   4. remove-double-neg N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 98.0

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

4. Applied egg-rr 98.0%

   \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{if}\;y \leq -1.9 \cdot 10^{-126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (/ z t) x)))
   (if (<= y -1.9e-126) t_1 (if (<= y 1.6e+27) (fma (/ z t) (- x) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (z / t), x);
	double tmp;
	if (y <= -1.9e-126) {
		tmp = t_1;
	} else if (y <= 1.6e+27) {
		tmp = fma((z / t), -x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(z / t), x)
	tmp = 0.0
	if (y <= -1.9e-126)
		tmp = t_1;
	elseif (y <= 1.6e+27)
		tmp = fma(Float64(z / t), Float64(-x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.9e-126], t$95$1, If[LessEqual[y, 1.6e+27], N[(N[(z / t), $MachinePrecision] * (-x) + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8999999999999999e-126 or 1.60000000000000008e27 < y

   1. Initial program 90.3%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto x + \frac{\color{blue}{z \cdot y}}{t} \]

      2. associate-/l* N/A

         \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]

      4. lower-/.f64 83.2

         \[\leadsto x + z \cdot \color{blue}{\frac{y}{t}} \]

   5. Simplified 83.2%

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

      1. lift-/.f64 N/A

         \[\leadsto x + z \cdot \color{blue}{\frac{y}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \frac{y}{t} + x} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{y}{t}} + x \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{t} \cdot z} + x \]

      6. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{t}} \cdot z + x \]

      7. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{y}{\frac{t}{z}}} + x \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\frac{t}{z}}} + x \]

      9. div-inv N/A

         \[\leadsto \color{blue}{y \cdot \frac{1}{\frac{t}{z}}} + x \]

      10. lift-/.f64 N/A

          \[\leadsto y \cdot \frac{1}{\color{blue}{\frac{t}{z}}} + x \]

      11. clear-num N/A

          \[\leadsto y \cdot \color{blue}{\frac{z}{t}} + x \]

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 88.3

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

   7. Applied egg-rr 88.3%

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

   if -1.8999999999999999e-126 < y < 1.60000000000000008e27

   1. Initial program 93.9%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 98.2

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

   4. Applied egg-rr 98.2%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 93.1

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

   7. Simplified 93.1%

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

Alternative 3: 84.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{-66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{x}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (/ z t) x)))
   (if (<= y -3.5e-66) t_1 (if (<= y 9e+26) (fma (- z) (/ x t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (z / t), x);
	double tmp;
	if (y <= -3.5e-66) {
		tmp = t_1;
	} else if (y <= 9e+26) {
		tmp = fma(-z, (x / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(z / t), x)
	tmp = 0.0
	if (y <= -3.5e-66)
		tmp = t_1;
	elseif (y <= 9e+26)
		tmp = fma(Float64(-z), Float64(x / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -3.5e-66], t$95$1, If[LessEqual[y, 9e+26], N[((-z) * N[(x / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.5e-66 or 8.99999999999999957e26 < y

   1. Initial program 89.4%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto x + \frac{\color{blue}{z \cdot y}}{t} \]

      2. associate-/l* N/A

         \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]

      4. lower-/.f64 85.1

         \[\leadsto x + z \cdot \color{blue}{\frac{y}{t}} \]

   5. Simplified 85.1%

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

      1. lift-/.f64 N/A

         \[\leadsto x + z \cdot \color{blue}{\frac{y}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \frac{y}{t} + x} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{y}{t}} + x \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{t} \cdot z} + x \]

      6. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{t}} \cdot z + x \]

      7. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{y}{\frac{t}{z}}} + x \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\frac{t}{z}}} + x \]

      9. div-inv N/A

         \[\leadsto \color{blue}{y \cdot \frac{1}{\frac{t}{z}}} + x \]

      10. lift-/.f64 N/A

          \[\leadsto y \cdot \frac{1}{\color{blue}{\frac{t}{z}}} + x \]

      11. clear-num N/A

          \[\leadsto y \cdot \color{blue}{\frac{z}{t}} + x \]

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 89.3

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

   7. Applied egg-rr 89.3%

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

   if -3.5e-66 < y < 8.99999999999999957e26

   1. Initial program 94.6%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 88.4

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

   5. Simplified 88.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 89.1

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

   7. Applied egg-rr 89.1%

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

Alternative 4: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{-56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+26}:\\ \;\;\;\;x - \frac{x \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (/ z t) x)))
   (if (<= y -1.05e-56) t_1 (if (<= y 7e+26) (- x (/ (* x z) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (z / t), x);
	double tmp;
	if (y <= -1.05e-56) {
		tmp = t_1;
	} else if (y <= 7e+26) {
		tmp = x - ((x * z) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(z / t), x)
	tmp = 0.0
	if (y <= -1.05e-56)
		tmp = t_1;
	elseif (y <= 7e+26)
		tmp = Float64(x - Float64(Float64(x * z) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.05e-56], t$95$1, If[LessEqual[y, 7e+26], N[(x - N[(N[(x * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05000000000000003e-56 or 6.9999999999999998e26 < y

   1. Initial program 89.2%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto x + \frac{\color{blue}{z \cdot y}}{t} \]

      2. associate-/l* N/A

         \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]

      4. lower-/.f64 84.9

         \[\leadsto x + z \cdot \color{blue}{\frac{y}{t}} \]

   5. Simplified 84.9%

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

      1. lift-/.f64 N/A

         \[\leadsto x + z \cdot \color{blue}{\frac{y}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \frac{y}{t} + x} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{y}{t}} + x \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{t} \cdot z} + x \]

      6. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{t}} \cdot z + x \]

      7. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{y}{\frac{t}{z}}} + x \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\frac{t}{z}}} + x \]

      9. div-inv N/A

         \[\leadsto \color{blue}{y \cdot \frac{1}{\frac{t}{z}}} + x \]

      10. lift-/.f64 N/A

          \[\leadsto y \cdot \frac{1}{\color{blue}{\frac{t}{z}}} + x \]

      11. clear-num N/A

          \[\leadsto y \cdot \color{blue}{\frac{z}{t}} + x \]

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 89.2

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

   7. Applied egg-rr 89.2%

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

   if -1.05000000000000003e-56 < y < 6.9999999999999998e26

   1. Initial program 94.6%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 88.6

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

   5. Simplified 88.6%

      \[\leadsto \color{blue}{x - \frac{z \cdot x}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+26}:\\ \;\;\;\;x - \frac{x \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{y - x}{t}\\ \mathbf{if}\;z \leq -6.2 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (/ (- y x) t))))
   (if (<= z -6.2e+37) t_1 (if (<= z 8e-17) (+ x (/ (* y z) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * ((y - x) / t);
	double tmp;
	if (z <= -6.2e+37) {
		tmp = t_1;
	} else if (z <= 8e-17) {
		tmp = x + ((y * z) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * ((y - x) / t)
    if (z <= (-6.2d+37)) then
        tmp = t_1
    else if (z <= 8d-17) then
        tmp = x + ((y * z) / t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = z * ((y - x) / t);
	double tmp;
	if (z <= -6.2e+37) {
		tmp = t_1;
	} else if (z <= 8e-17) {
		tmp = x + ((y * z) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * ((y - x) / t)
	tmp = 0
	if z <= -6.2e+37:
		tmp = t_1
	elif z <= 8e-17:
		tmp = x + ((y * z) / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(Float64(y - x) / t))
	tmp = 0.0
	if (z <= -6.2e+37)
		tmp = t_1;
	elseif (z <= 8e-17)
		tmp = Float64(x + Float64(Float64(y * z) / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * ((y - x) / t);
	tmp = 0.0;
	if (z <= -6.2e+37)
		tmp = t_1;
	elseif (z <= 8e-17)
		tmp = x + ((y * z) / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.2e+37], t$95$1, If[LessEqual[z, 8e-17], N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.2000000000000004e37 or 8.00000000000000057e-17 < z

   1. Initial program 84.9%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 86.7

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

   5. Simplified 86.7%

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

   if -6.2000000000000004e37 < z < 8.00000000000000057e-17

   1. Initial program 97.9%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 88.6

         \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]

   5. Simplified 88.6%

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

Alternative 6: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{y - x}{t}\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (/ (- y x) t))))
   (if (<= z -1.7e+40) t_1 (if (<= z 1.05e-13) (fma y (/ z t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * ((y - x) / t);
	double tmp;
	if (z <= -1.7e+40) {
		tmp = t_1;
	} else if (z <= 1.05e-13) {
		tmp = fma(y, (z / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(Float64(y - x) / t))
	tmp = 0.0
	if (z <= -1.7e+40)
		tmp = t_1;
	elseif (z <= 1.05e-13)
		tmp = fma(y, Float64(z / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.7e+40], t$95$1, If[LessEqual[z, 1.05e-13], N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.69999999999999994e40 or 1.04999999999999994e-13 < z

   1. Initial program 84.9%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 86.7

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

   5. Simplified 86.7%

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

   if -1.69999999999999994e40 < z < 1.04999999999999994e-13

   1. Initial program 97.9%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto x + \frac{\color{blue}{z \cdot y}}{t} \]

      2. associate-/l* N/A

         \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]

      4. lower-/.f64 83.9

         \[\leadsto x + z \cdot \color{blue}{\frac{y}{t}} \]

   5. Simplified 83.9%

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

      1. lift-/.f64 N/A

         \[\leadsto x + z \cdot \color{blue}{\frac{y}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \frac{y}{t} + x} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{y}{t}} + x \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{t} \cdot z} + x \]

      6. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{t}} \cdot z + x \]

      7. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{y}{\frac{t}{z}}} + x \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\frac{t}{z}}} + x \]

      9. div-inv N/A

         \[\leadsto \color{blue}{y \cdot \frac{1}{\frac{t}{z}}} + x \]

      10. lift-/.f64 N/A

          \[\leadsto y \cdot \frac{1}{\color{blue}{\frac{t}{z}}} + x \]

      11. clear-num N/A

          \[\leadsto y \cdot \color{blue}{\frac{z}{t}} + x \]

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 88.1

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

   7. Applied egg-rr 88.1%

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

Alternative 7: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{-218}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-261}:\\ \;\;\;\;-\frac{x \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (/ z t) x)))
   (if (<= t -3.8e-218) t_1 (if (<= t 5.6e-261) (- (/ (* x z) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (z / t), x);
	double tmp;
	if (t <= -3.8e-218) {
		tmp = t_1;
	} else if (t <= 5.6e-261) {
		tmp = -((x * z) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(z / t), x)
	tmp = 0.0
	if (t <= -3.8e-218)
		tmp = t_1;
	elseif (t <= 5.6e-261)
		tmp = Float64(-Float64(Float64(x * z) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -3.8e-218], t$95$1, If[LessEqual[t, 5.6e-261], (-N[(N[(x * z), $MachinePrecision] / t), $MachinePrecision]), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.7999999999999999e-218 or 5.60000000000000018e-261 < t

   1. Initial program 91.0%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto x + \frac{\color{blue}{z \cdot y}}{t} \]

      2. associate-/l* N/A

         \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]

      4. lower-/.f64 80.3

         \[\leadsto x + z \cdot \color{blue}{\frac{y}{t}} \]

   5. Simplified 80.3%

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

      1. lift-/.f64 N/A

         \[\leadsto x + z \cdot \color{blue}{\frac{y}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \frac{y}{t} + x} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{y}{t}} + x \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{t} \cdot z} + x \]

      6. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{t}} \cdot z + x \]

      7. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{y}{\frac{t}{z}}} + x \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\frac{t}{z}}} + x \]

      9. div-inv N/A

         \[\leadsto \color{blue}{y \cdot \frac{1}{\frac{t}{z}}} + x \]

      10. lift-/.f64 N/A

          \[\leadsto y \cdot \frac{1}{\color{blue}{\frac{t}{z}}} + x \]

      11. clear-num N/A

          \[\leadsto y \cdot \color{blue}{\frac{z}{t}} + x \]

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 82.3

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

   7. Applied egg-rr 82.3%

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

   if -3.7999999999999999e-218 < t < 5.60000000000000018e-261

   1. Initial program 99.9%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 80.6

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

   5. Simplified 80.6%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 79.7

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

   8. Simplified 79.7%

      \[\leadsto \color{blue}{\frac{x \cdot z}{-t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-218}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-261}:\\ \;\;\;\;-\frac{x \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.8%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 97.7

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

4. Applied egg-rr 97.7%

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

Alternative 9: 76.1% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x y z t) :precision binary64 (fma y (/ z t) x))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.8%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

   1. *-commutative N/A

      \[\leadsto x + \frac{\color{blue}{z \cdot y}}{t} \]

   2. associate-/l* N/A

      \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]

   3. lower-*.f64 N/A

      \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]

   4. lower-/.f64 74.6

      \[\leadsto x + z \cdot \color{blue}{\frac{y}{t}} \]

5. Simplified 74.6%

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

   1. lift-/.f64 N/A

      \[\leadsto x + z \cdot \color{blue}{\frac{y}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto x + \color{blue}{z \cdot \frac{y}{t}} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{z \cdot \frac{y}{t} + x} \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{z \cdot \frac{y}{t}} + x \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\frac{y}{t} \cdot z} + x \]

   6. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y}{t}} \cdot z + x \]

   7. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{y}{\frac{t}{z}}} + x \]

   8. lift-/.f64 N/A

      \[\leadsto \frac{y}{\color{blue}{\frac{t}{z}}} + x \]

   9. div-inv N/A

      \[\leadsto \color{blue}{y \cdot \frac{1}{\frac{t}{z}}} + x \]

   10. lift-/.f64 N/A

       \[\leadsto y \cdot \frac{1}{\color{blue}{\frac{t}{z}}} + x \]

   11. clear-num N/A

       \[\leadsto y \cdot \color{blue}{\frac{z}{t}} + x \]

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 78.6

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

7. Applied egg-rr 78.6%

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

Alternative 10: 40.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ y \cdot \frac{z}{t} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.8%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{z \cdot y}}{t} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{z \cdot \frac{y}{t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{z \cdot \frac{y}{t}} \]

   4. lower-/.f64 33.6

      \[\leadsto z \cdot \color{blue}{\frac{y}{t}} \]

5. Simplified 33.6%

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

   1. lift-/.f64 N/A

      \[\leadsto z \cdot \color{blue}{\frac{y}{t}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{y}{t} \cdot z} \]

   3. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y}{t}} \cdot z \]

   4. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{y}{\frac{t}{z}}} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{y}{\color{blue}{\frac{t}{z}}} \]

   6. div-inv N/A

      \[\leadsto \color{blue}{y \cdot \frac{1}{\frac{t}{z}}} \]

   7. lift-/.f64 N/A

      \[\leadsto y \cdot \frac{1}{\color{blue}{\frac{t}{z}}} \]

   8. clear-num N/A

      \[\leadsto y \cdot \color{blue}{\frac{z}{t}} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]

   10. lower-/.f64 38.0

       \[\leadsto y \cdot \color{blue}{\frac{z}{t}} \]

7. Applied egg-rr 38.0%

   \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]
8. Add Preprocessing

Developer Target 1: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x < -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x < 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (< x -9.025511195533005e-135)
   (- x (* (/ z t) (- x y)))
   (if (< x 4.275032163700715e-250)
     (+ x (* (/ (- y x) t) z))
     (+ x (/ (- y x) (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x < -9.025511195533005e-135) {
		tmp = x - ((z / t) * (x - y));
	} else if (x < 4.275032163700715e-250) {
		tmp = x + (((y - x) / t) * z);
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x < (-9.025511195533005d-135)) then
        tmp = x - ((z / t) * (x - y))
    else if (x < 4.275032163700715d-250) then
        tmp = x + (((y - x) / t) * z)
    else
        tmp = x + ((y - x) / (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x < -9.025511195533005e-135) {
		tmp = x - ((z / t) * (x - y));
	} else if (x < 4.275032163700715e-250) {
		tmp = x + (((y - x) / t) * z);
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x < -9.025511195533005e-135:
		tmp = x - ((z / t) * (x - y))
	elif x < 4.275032163700715e-250:
		tmp = x + (((y - x) / t) * z)
	else:
		tmp = x + ((y - x) / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x < -9.025511195533005e-135)
		tmp = Float64(x - Float64(Float64(z / t) * Float64(x - y)));
	elseif (x < 4.275032163700715e-250)
		tmp = Float64(x + Float64(Float64(Float64(y - x) / t) * z));
	else
		tmp = Float64(x + Float64(Float64(y - x) / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x < -9.025511195533005e-135)
		tmp = x - ((z / t) * (x - y));
	elseif (x < 4.275032163700715e-250)
		tmp = x + (((y - x) / t) * z);
	else
		tmp = x + ((y - x) / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Less[x, -9.025511195533005e-135], N[(x - N[(N[(z / t), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[x, 4.275032163700715e-250], N[(x + N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y - x), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -1805102239106601/200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- x (* (/ z t) (- x y))) (if (< x 855006432740143/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z))))))

  (+ x (/ (* (- y x) z) t)))